1926:
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464:
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2572:
2924:
2099:
40:
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212:
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Several other shapes can be defined from a set of points in a similar way to the convex hull, as the minimal superset with some property, the intersection of all shapes containing the points from a given family of shapes, or the union of all combinations of points for a certain type of combination.
915:
In two dimensions, the convex hull is sometimes partitioned into two parts, the upper hull and the lower hull, stretching between the leftmost and rightmost points of the hull. More generally, for convex hulls in any dimension, one can partition the boundary of the hull into upward-facing points
1319:
of a convex set is a point in the set that does not lie on any open line segment between any other two points of the same set. For a convex hull, every extreme point must be part of the given set, because otherwise it cannot be formed as a convex combination of given points. According to the
3343:, the term "convex hull" had become standard; Dines adds that he finds the term unfortunate, because the colloquial meaning of the word "hull" would suggest that it refers to the surface of a shape, whereas the convex hull includes the interior and not just the surface.
2830:
of a finite point set give a nested family of (non-convex) geometric objects describing the shape of a point set at different levels of detail. Each of alpha shape is the union of some of the features of the
Delaunay triangulation, selected by comparing their
3221:
can make the minimum convex polygon excessively large, which has motivated relaxed approaches that contain only a subset of the observations, for instance by choosing one of the convex layers that is close to a target percentage of the samples, or in the
3298:
of a material, only those measurements on the lower convex hull will be stable. When removing a point from the hull and then calculating its distance to the hull, its distance to the new hull represents the degree of stability of the phase.
1328:) is the convex hull of its extreme points. However, this may not be true for convex sets that are not compact; for instance, the whole Euclidean plane and the open unit ball are both convex, but neither one has any extreme points.
1273:
2531:
structures can keep track of the convex hull for points moving continuously. The construction of convex hulls also serves as a tool, a building block for a number of other computational-geometric algorithms such as the
916:(points for which an upward ray is disjoint from the hull), downward-facing points, and extreme points. For three-dimensional hulls, the upward-facing and downward-facing parts of the boundary form topological disks.
3011:, points that do not belong to the hyperbolic space itself but lie on the boundary of a model of that space. The boundaries of convex hulls of ideal points of three-dimensional hyperbolic space are analogous to
7018:
4688:
2713:
or rectilinear convex hull is the intersection of all orthogonally convex and connected supersets, where a set is orthogonally convex if it contains all axis-parallel segments between pairs of its points.
2839:
of a point set are a nested family of convex polygons, the outermost of which is the convex hull, with the inner layers constructed recursively from the points that are not vertices of the convex hull.
1601:
commute with each other, in the sense that the
Minkowski sum of convex hulls of sets gives the same result as the convex hull of the Minkowski sum of the same sets. This provides a step towards the
2246:, a number of algorithms are known for computing the convex hull for a finite set of points and for other geometric objects. Computing the convex hull means constructing an unambiguous, efficient
1189:
is always itself open, and the convex hull of a compact set is always itself compact. However, there exist closed sets for which the convex hull is not closed. For instance, the closed set
1617:
operation to constructing the convex hull of a set of points is constructing the intersection of a family of closed halfspaces that all contain the origin (or any other designated point).
2515:
1889:
2027:
1677:
5597:
Kim, Sooran; Kim, Kyoo; Koo, Jahyun; Lee, Hoonkyung; Min, Byung Il; Kim, Duck Young (December 2019), "Pressure-induced phase transitions and superconductivity in magnesium carbides",
7011:
1048:), then it equals the closed convex hull. However, an intersection of closed half-spaces is itself closed, so when a convex hull is not closed it cannot be represented in this way.
3186:
is that it lies within the convex hull of its control points. This so-called "convex hull property" can be used, for instance, in quickly detecting intersections of these curves.
2911:
helps find their crossings, and convex hulls are part of the measurement of boat hulls. And in the study of animal behavior, convex hulls are used in a standard definition of the
2824:
2786:
1740:
2088:
1470:
3116:
of solutions to a combinatorial problem. If the facets of these polytopes can be found, describing the polytopes as intersections of halfspaces, then algorithms based on
2656:
intersects the object. Equivalently it is the intersection of the (non-convex) cones generated by the outline of the object with respect to each viewpoint. It is used in
183:. Convex hulls have wide applications in mathematics, statistics, combinatorial optimization, economics, geometric modeling, and ethology. Related structures include the
7004:
6522:
5693:
4593:
2431:
2389:
158:
2689:
6840:
6122:
5133:
4494:
1589:, the shelling antimatroid of the point set. Every antimatroid can be represented in this way by convex hulls of points in a Euclidean space of high-enough dimension.
3085:
form a nested family of convex sets, with the convex hull outermost, and the bagplot also displays another polygon from this nested family, the contour of 50% depth.
2047:
1987:
2465:
2220:. The definition can be extended to the convex hull of a set of functions (obtained from the convex hull of the union of their epigraphs, or equivalently from their
5579:
Kernohan, Brian J.; Gitzen, Robert A.; Millspaugh, Joshua J. (2001), "Analysis of animal space use and movements", in
Millspaugh, Joshua; Marzluff, John M. (eds.),
5422:
Hautier, Geoffroy (2014), "Data mining approaches to high-throughput crystal structure and compound prediction", in Atahan-Evrenk, Sule; Aspuru-Guzik, Alan (eds.),
857:
1707:
805:
428:
of the points encloses them with arbitrarily small surface area, smaller than the surface area of the convex hull. However, in higher dimensions, variants of the
1159:
1112:
5355:
2654:
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2351:
2327:
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2194:
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1132:
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1038:
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994:
970:
905:
885:
825:
779:
759:
739:
712:
692:
672:
652:
632:
609:
589:
569:
549:
529:
509:
489:
422:
402:
378:
343:
317:
291:
269:
245:
6084:
Nilsen, Erlend B.; Pedersen, Simen; Linnell, John D. C. (2008), "Can minimum convex polygon home ranges be used to draw biologically meaningful conclusions?",
4412:
3003:
The definitions of a convex set as containing line segments between its points, and of a convex hull as the intersection of all convex supersets, apply to
7208:
1746:, and (by the Krein–Milman theorem) every convex polytope is the convex hull of its vertices. It is the unique convex polytope whose vertices belong to
718:
1195:
6347:
3170:
can be used to show that, for large markets, this approximation is accurate, and leads to a "quasi-equilibrium" for the original non-convex market.
424:. This formulation does not immediately generalize to higher dimensions: for a finite set of points in three-dimensional space, a neighborhood of a
1949:. Reflecting a pocket across its convex hull edge expands the given simple polygon into a polygon with the same perimeter and larger area, and the
7108:
6876:
Williams, Jason; Rossignac, Jarek (2005), "Tightening: curvature-limiting morphological simplification", in
Kobbelt, Leif; Shapiro, Vadim (eds.),
5021:
4753:
1325:
2945:
of multivariate polynomials are convex hulls of points derived from the exponents of the terms in the polynomial, and can be used to analyze the
7200:
6357:
6266:
6075:
5439:
4919:
2835:
to the parameter alpha. The point set itself forms one endpoint of this family of shapes, and its convex hull forms the other endpoint. The
4793:
4718:
2595:
is the smallest affine subspace of a
Euclidean space containing a given set, or the union of all affine combinations of points in the set.
6532:
5500:
2266:
of the hull. In two dimensions, it may suffice more simply to list the points that are vertices, in their cyclic order around the hull.
2602:
is the smallest linear subspace of a vector space containing a given set, or the union of all linear combinations of points in the set.
2029:, there will be times during the Brownian motion where the moving particle touches the boundary of the convex hull at a point of angle
7213:
2927:
Partition of seven points into three subsets with intersecting convex hulls, guaranteed to exist for any seven points in the plane by
2131:
1937:
encloses the given polygon and is partitioned by it into regions, one of which is the polygon itself. The other regions, bounded by a
6499:
6229:
5941:
5745:
5588:
5530:
Kashiwabara, Kenji; Nakamura, Masataka; Okamoto, Yoshio (2005), "The affine representation theorem for abstract convex geometries",
5344:
5205:
5122:
2702:
is the intersection of all relatively convex supersets, where a set within the same polygon is relatively convex if it contains the
2527:
data structures can be used to keep track of the convex hull of a set of points undergoing insertions and deletions of points, and
2517:, matching the worst-case output complexity of the problem. The convex hull of a simple polygon in the plane can be constructed in
2438:
1332:
extends this theory from finite convex combinations of extreme points to infinite combinations (integrals) in more general spaces.
5386:
5532:
4457:
2114:
or finite set of space curves in general position in three-dimensional space, the parts of the boundary away from the curves are
7233:
5463:(1992), "Hyperconvex hulls of metric spaces", Proceedings of the Symposium on General Topology and Applications (Oxford, 1989),
2250:
of the required convex shape. Output representations that have been considered for convex hulls of point sets include a list of
2130:, the convex hull of two semicircles in perpendicular planes with a common center, and D-forms, the convex shapes obtained from
7462:
7150:
1920:
5727:
3251:
2269:
For convex hulls in two or three dimensions, the complexity of the corresponding algorithms is usually estimated in terms of
1945:. Computing the same decomposition recursively for each pocket forms a hierarchical description of a given polygon called its
110:
problems of finding the convex hull of a finite set of points in the plane or other low-dimensional
Euclidean spaces, and its
71:
that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a
6990:
6615:
6188:
6014:
5645:
4631:
4360:
4895:
6878:
Proceedings of the Tenth ACM Symposium on Solid and
Physical Modeling 2005, Cambridge, Massachusetts, USA, June 13-15, 2005
6275:
Rappoport, Ari (1992), "An efficient adaptive algorithm for constructing the convex differences tree of a simple polygon",
5056:, London Mathematical Society Lecture Note Series, vol. 111, Cambridge: Cambridge University Press, pp. 113–253,
83:
subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.
6956:
6756:
6751:
6305:
5465:
4941:
3227:
3007:
as well as to
Euclidean spaces. However, in hyperbolic space, it is also possible to consider the convex hulls of sets of
3128:
of weights by finding and checking each convex hull vertex, often more efficiently than checking all possible solutions.
3124:, a different type of convex hull is also used, the convex hull of the weight vectors of solutions. One can maximize any
7467:
3247:
3121:
3167:
471:
It is not obvious that the first definition makes sense: why should there exist a unique minimal convex set containing
6951:
4906:, Contemporary Mathematics, vol. 453, Providence, Rhode Island: American Mathematical Society, pp. 231–255,
2732:
7218:
5270:(1873), "A method of geometrical representation of the thermodynamic properties of substances by means of surfaces",
3213:, the study of animal behavior, where it is a classic, though perhaps simplistic, approach in estimating an animal's
6251:
Mathematical
Programming: The State of the Art (XIth International Symposium on Mathematical Programming, Bonn 1982)
7452:
6457:
3246:
of any quantum system — the set of all ways the system can be prepared — is a convex hull whose extreme points are
3105:
2949:
behavior of the polynomial and the valuations of its roots. Convex hulls and polynomials also come together in the
2896:
2392:
7248:
1950:
1602:
1290:
435:
For objects in three dimensions, the first definition states that the convex hull is the smallest possible convex
6788:
6435:
5786:
3163:
3016:
2736:
7238:
2470:
1844:
1321:
1310:
7457:
7223:
7065:
5223:
3109:
3093:
3027:
3020:
2900:
2888:
2609:
or positive hull of a subset of a vector space is the set of all positive combinations of points in the subset.
1992:
1647:
1289:
The compactness of convex hulls of compact sets, in finite-dimensional
Euclidean spaces, is generalized by the
30:
This article is about the smallest convex shape enclosing a given shape. For boats whose hulls are convex, see
3294:(1873), although the paper was published before the convex hull was so named. In a set of energies of several
2950:
2663:
The circular hull or alpha-hull of a subset of the plane is the intersection of all disks with a given radius
1177:. The points on or above the red curve provide an example of a closed set whose convex hull is open (the open
7308:
7285:
7179:
7103:
3254:
proves that any mixed state can in fact be written as a convex combination of pure states in multiple ways.
160:
for two or three dimensional point sets, and in time matching the worst-case output complexity given by the
3143:
591:
is included among the sets being intersected. Thus, it is exactly the unique minimal convex set containing
7426:
7387:
7303:
7228:
7155:
7140:
7093:
6726:
6366:
6246:
5541:
4802:
4551:
3193:
is a measurement of the size of a sailing vessel, defined using the convex hull of a cross-section of the
3159:
3137:
2904:
2748:
2721:
2710:
2563:
2243:
2237:
2173:
440:
192:
184:
180:
119:
6486:, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge: Cambridge University Press,
3069:. The outer shaded region is the convex hull, and the inner shaded region is the 50% Tukey depth contour.
2977:
2791:
2329:. For higher-dimensional hulls, the number of faces of other dimensions may also come into the analysis.
7447:
7160:
6390:
5866:
5495:
5302:
4853:
4542:
3307:
The lower convex hull of points in the plane appears, in the form of a Newton polygon, in a letter from
2740:
2149:
2143:
973:
115:
6742:(1986), "An optimal algorithm for computing the relative convex hull of a set of points in a polygon",
2997:
2928:
2859:
Convex hulls have wide applications in many fields. Within mathematics, convex hulls are used to study
2126:, the convex hull of two circles in perpendicular planes, each passing through the other's center, the
654:, so the set of all convex combinations is contained in the intersection of all convex sets containing
4785:
2762:
1716:
7165:
7031:
6910:
6093:
6046:
5664:
5606:
5267:
5240:
5048:(1987), "Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces", in
5012:
5008:
3291:
3125:
2695:
2577:
2056:
1966:
867:
and in three-dimensional space it is a tetrahedron. Therefore, every convex combination of points of
404:
and then releasing it, allowing it to contract; when it becomes taut, it encloses the convex hull of
6731:
4851:
Cranston, M.; Hsu, P.; March, P. (1989), "Smoothness of the convex hull of planar Brownian motion",
4807:
4556:
3339:). Other terms, such as "convex envelope", were also used in this time frame. By 1938, according to
432:
of finding a minimum-energy surface above a given shape can have the convex hull as their solution.
227:
if it contains the line segments connecting each pair of its points. The convex hull of a given set
7098:
7088:
7083:
6658:
5546:
5426:, Topics in Current Chemistry, vol. 345, Springer International Publishing, pp. 139–179,
5049:
5041:
3147:
3050:
3031:
2946:
2528:
2524:
2434:
2115:
2050:
1925:
1798:
1791:
942:
353:
161:
1965:
in the plane, at any fixed time, has probability 1 of having a convex hull whose boundary forms a
1449:
443:, and the definition using convex combinations may be extended from Euclidean spaces to arbitrary
7327:
7045:
6946:
6927:
6897:
6857:
6827:
6739:
6718:
6683:
6642:
6567:
6541:
6407:
6332:
6292:
6249:(1983), "Polyhedral combinatorics", in Bachem, Achim; Korte, Bernhard; Grötschel, Martin (eds.),
6109:
5986:
5893:
5839:
5803:
5759:
5680:
5654:
5287:
5256:
5166:
5150:
5092:
4958:
4899:
4872:
4820:
4648:
4610:
4577:
4533:
4513:
4437:
3201:, the perimeter of the cross-section itself, except for boats and ships that have a convex hull.
3179:
3155:
3117:
3035:
2957:
of the derivative of a polynomial all lie within the convex hull of the roots of the polynomial.
2954:
2752:
934:
296:
111:
76:
6120:
Oberman, Adam M. (2007), "The convex envelope is the solution of a nonlinear obstacle problem",
5330:
5224:"A local nearest-neighbor convex-hull construction of home ranges and utilization distributions"
2993:
2398:
2356:
125:
3000:
concern the existence of partitions of point sets into subsets with intersecting convex hulls.
2666:
7145:
6986:
6965:
6606:
6516:
6495:
6353:
6262:
6225:
6071:
5998:
5937:
5781:
5741:
5632:
5584:
5454:
5445:
5435:
5340:
5201:
5185:
5118:
5069:
4915:
4485:
3332:
3223:
3194:
3074:
2989:
2876:
2872:
2657:
2533:
2251:
2221:
1743:
1402:
946:
444:
31:
6209:
5104:
4404:
4384:
4348:
3482:, p. 6. The idea of partitioning the hull into two chains comes from an efficient variant of
2032:
1972:
7298:
7243:
7134:
7129:
6919:
6889:
6881:
6849:
6797:
6765:
6667:
6624:
6596:
6551:
6487:
6466:
6444:
6399:
6373:, Princeton Mathematical Series, vol. 28, Princeton, N.J.: Princeton University Press,
6314:
6284:
6254:
6217:
6197:
6161:
6131:
6101:
6063:
6023:
5970:
5921:
5883:
5875:
5848:
5816:
5795:
5733:
5702:
5672:
5622:
5614:
5551:
5509:
5474:
5427:
5364:
5334:
5311:
5248:
5193:
5181:
5142:
5110:
5084:
5030:
4978:
4950:
4907:
4862:
4840:
4812:
4781:
4762:
4748:
4727:
4697:
4663:
4640:
4602:
4588:
4561:
4537:
4503:
4466:
4421:
4369:
3364:
3316:
3113:
3004:
2892:
2759:, are mathematically related to convex hulls: the Delaunay triangulation of a point set in
2717:
2444:
2259:
2255:
2225:
1787:
1346:
1283:
1178:
452:
429:
99:
6996:
6869:
6811:
6779:
6711:
6679:
6638:
6587:
6563:
6509:
6419:
6378:
6328:
6239:
6175:
6145:
6035:
5982:
5951:
5912:
Laurentini, A. (1994), "The visual hull concept for silhouette-based image understanding",
5905:
5864:; Ĺ mulian, V. (1940), "On regularly convex sets in the space conjugate to a Banach space",
5755:
5716:
5563:
5523:
5488:
5401:
5378:
5323:
5215:
5162:
5061:
5001:
4982:
4970:
4929:
4884:
4774:
4741:
4709:
4622:
4573:
4525:
4478:
4433:
4396:
2616:
of a three-dimensional object, with respect to a set of viewpoints, consists of the points
830:
7318:
7289:
7263:
7258:
7253:
7184:
7169:
7078:
7050:
7027:
6865:
6807:
6775:
6707:
6675:
6634:
6583:
6559:
6505:
6426:
6415:
6374:
6343:
6324:
6235:
6171:
6141:
6031:
5978:
5947:
5901:
5820:
5812:
5751:
5712:
5559:
5519:
5484:
5397:
5374:
5319:
5211:
5158:
5057:
4997:
4966:
4925:
4880:
4770:
4737:
4705:
4683:
4618:
4569:
4521:
4474:
4429:
4392:
3328:
3324:
3312:
3262:
3239:
3089:
2981:
2965:
2961:
2942:
2868:
2848:
2756:
2197:
1962:
1938:
1710:
1639:
1614:
1279:
1174:
436:
220:
196:
172:
72:
4716:
Chang, J. S.; Yap, C.-K. (1986), "A polynomial solution for the potato-peeling problem",
4161:. See in particular Section 16.9, Non Convexity and Approximate Equilibrium, pp. 209–210.
3183:
2908:
1686:
784:
6097:
5668:
5610:
5394:
Optimizing methods in statistics (Proc. Sympos., Ohio State Univ., Columbus, Ohio, 1971)
5282:
5244:
4358:
Andrew, A. M. (1979), "Another efficient algorithm for convex hulls in two dimensions",
3081:, a method for visualizing the spread of two-dimensional sample points. The contours of
1141:
1094:
634:
must (by the assumption that it is convex) contain all convex combinations of points in
7405:
7313:
7174:
7073:
6653:
6481:
6477:
5627:
5572:
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
5460:
5405:
5177:
5016:
4671:
4452:
3287:
3024:
2934:
2699:
2639:
2619:
2557:
2336:
2312:
2292:
2272:
2247:
2203:
2179:
2155:
1934:
1894:
1824:
1804:
1769:
1749:
1680:
1564:
1544:
1524:
1495:
1475:
1429:
1409:
1380:
1360:
1329:
1135:
1117:
1074:
1054:
1023:
999:
979:
955:
890:
870:
810:
764:
744:
724:
697:
677:
657:
637:
617:
594:
574:
554:
534:
514:
494:
474:
407:
387:
363:
328:
302:
276:
254:
230:
168:
4844:
4508:
4470:
3096:
is the convex hull of the risk points of its underlying deterministic decision rules.
2716:
The orthogonal convex hull is a special case of a much more general construction, the
1891:. In particular, in two and three dimensions the number of faces is at most linear in
7441:
7372:
7364:
7360:
7356:
7352:
7348:
7189:
6802:
6770:
6687:
6470:
6336:
6216:, Algorithms and Computation in Mathematics, vol. 11, Springer, pp. 12–13,
6201:
6027:
5807:
5514:
5479:
5315:
5293:
5252:
5170:
5096:
5045:
4667:
4644:
4489:
4373:
3295:
3012:
2969:
2836:
2119:
1598:
1316:
1298:
1091:-dimensional, then every point of the hull belongs to an open convex hull of at most
439:
of the objects. The definition using intersections of convex sets may be extended to
425:
188:
95:
6901:
6646:
6571:
6296:
6113:
5990:
5684:
5369:
5260:
4824:
4652:
4441:
3400:
3398:
3396:
2571:
761:-dimensional Euclidean space, every convex combination of finitely many points from
352:
in the Euclidean plane, not all on one line, the boundary of the convex hull is the
7410:
6696:"Fixed points for condensing multifunctions in metric spaces with convex structure"
6448:
6042:
5958:
5830:
5767:
5763:
5723:
4891:
4659:
4581:
3424:
3308:
3162:
can be used to prove the existence of an equilibrium. When actual economic data is
2847:
of a polygon is the largest convex polygon contained inside it. It can be found in
2844:
2832:
2725:
2606:
2309:, the number of points on the convex hull, which may be significantly smaller than
2263:
2134:
for a surface formed by gluing together two planar convex sets of equal perimeter.
1294:
674:. Conversely, the set of all convex combinations is itself a convex set containing
448:
200:
106:
can be represented by applying this closure operator to finite sets of points. The
6695:
6610:
6555:
6136:
5707:
6258:
6067:
5732:, Lecture Notes in Computer Science, vol. 606, Heidelberg: Springer-Verlag,
5555:
5353:
Gustin, William (1947), "On the interior of the convex hull of a Euclidean set",
2660:
as the largest shape that could have the same outlines from the given viewpoints.
511:? However, the second definition, the intersection of all convex sets containing
7400:
7395:
6982:
6969:
6744:
Proceedings of EURASIP, Signal Processing III: Theories and Applications, Part 2
6183:
5297:
4936:
3483:
3340:
3243:
3190:
3082:
3008:
2923:
2880:
2827:
2613:
2599:
2592:
2518:
2330:
2111:
1586:
1517:
1268:{\displaystyle \left\{(x,y)\mathop {\bigg |} y\geq {\frac {1}{1+x^{2}}}\right\}}
1045:
381:
349:
176:
103:
91:
80:
6530:
Seaton, Katherine A. (2017), "Sphericons and D-forms: a crocheted connection",
5618:
4911:
7323:
7293:
7055:
6430:
6385:
6288:
6221:
6105:
6012:(1979), "A linear algorithm for finding the convex hull of a simple polygon",
6009:
5888:
5861:
5826:
5691:
Kiselman, Christer O. (2002), "A semigroup of operators in convexity theory",
5676:
5197:
4701:
4448:
4380:
3214:
3198:
3151:
3015:
in Euclidean space, and their metric properties play an important role in the
2985:
2973:
2938:
2912:
2864:
2860:
2098:
1134:. The sets of vertices of a square, regular octahedron, or higher-dimensional
1041:
1017:
463:
224:
68:
44:
6908:
Worton, Bruce J. (1995), "A convex hull-based estimator of home-range size",
6491:
5034:
4867:
4766:
6974:
6885:
6671:
6629:
5231:
5131:
Gardner, L. Terrell (1984), "An elementary proof of the Russo-Dye theorem",
5054:
Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984)
3923:
3921:
3267:
3250:
known as pure states and whose interior points are called mixed states. The
3023:. Hyperbolic convex hulls have also been used as part of the calculation of
2127:
1293:, according to which the closed convex hull of a weakly compact subset of a
357:
107:
39:
5853:
5636:
5449:
4565:
1585:
When applied to a finite set of points, this is the closure operator of an
17:
5737:
5424:
Prediction and Calculation of Crystal Structures: Methods and Applications
5114:
4831:
Chen, Qinyu; Wang, Guozhao (March 2003), "A class of BĂ©zier-like curves",
167:
As well as for finite point sets, convex hulls have also been studied for
7332:
6656:(1914), "Bedingt konvergente Reihen und konvexe Systeme. (Fortsetzung)",
6186:(1984), "On the definition and computation of rectilinear convex hulls",
5659:
5431:
5339:, Graduate Texts in Mathematics, vol. 221 (2nd ed.), Springer,
3210:
2703:
2541:
1186:
864:
87:
52:
6831:
6754:(1993), "Convex hulls and isometries of cusped hyperbolic 3-manifolds",
5192:, Mathematics: Theory & Applications, Birkhäuser, pp. 193–213,
3420:
3045:
for the application of convex hulls to this subject, and the section on
1630:
1324:, every compact convex set in a Euclidean space (or more generally in a
1169:
7113:
6931:
6861:
6600:
6578:
Sedykh, V. D. (1981), "Structure of the convex hull of a space curve",
6411:
6319:
6303:
Reay, John R. (1979), "Several generalizations of Tverberg's theorem",
5974:
5897:
5799:
5154:
4962:
4876:
4816:
4732:
4614:
4517:
3315:
in 1676. The term "convex hull" itself appears as early as the work of
3218:
3078:
3066:
2907:
to non-convex markets. In geometric modeling, the convex hull property
2884:
860:
322:
211:
6166:
5925:
6893:
5088:
4425:
3827:
3811:
3691:
3679:
3655:
3479:
3404:
3271:
694:, so it also contains the intersection of all convex sets containing
6923:
6853:
6403:
6388:(1961), "Holomorphically convex sets in several complex variables",
5879:
5146:
4954:
4606:
6546:
3715:
3209:
The convex hull is commonly known as the minimum convex polygon in
3061:
4591:(1935), "Integration of functions with values in a Banach space",
3261:
3060:
2922:
2537:
2123:
2103:
2097:
1924:
1629:
1168:
462:
451:; convex hulls may also be generalized in a more abstract way, to
210:
38:
6721:(1983), "Solving geometric problems with the rotating calipers",
5570:
Katoh, Naoki (1992), "Bicriteria network optimization problems",
5283:
The Scientific Papers of J. Willard Gibbs, Vol. I: Thermodynamics
714:, and therefore the second and third definitions are equivalent.
4689:
International Journal of Computational Geometry and Applications
1345:
The convex-hull operator has the characteristic properties of a
94:
are compact. Every compact convex set is the convex hull of its
7000:
5834:
2887:
visualization of two-dimensional data, and define risk sets of
1929:
Convex hull ( in blue and yellow) of a simple polygon (in blue)
1605:
bounding the distance of a Minkowski sum from its convex hull.
5914:
IEEE Transactions on Pattern Analysis and Machine Intelligence
4027:
1597:
The operations of constructing the convex hull and taking the
6058:
Nicola, Piercarlo (2000), "General Competitive Equilibrium",
3927:
3367:. However, this term is also frequently used to refer to the
2903:. In economics, convex hulls can be used to apply methods of
5272:
Transactions of the Connecticut Academy of Arts and Sciences
5190:
Discriminants, Resultants, and Multidimensional Determinants
3951:
2053:
of this set of exceptional times is (with high probability)
531:, is well-defined. It is a subset of every other convex set
4629:
Brown, K. Q. (1979), "Voronoi diagrams from convex hulls",
3383:
3381:
3282:
is expected to be unstable as it lies above the lower hull.
2391:. For points in two and three dimensions, more complicated
5963:
International Journal of Computer and Information Sciences
5961:(1983), "On finding the convex hull of a simple polygon",
5643:
Kirkpatrick, K. A. (2006), "The Schrödinger–HJW theorem",
3627:
3077:, the convex hull provides one of the key components of a
1953:
states that this expansion process eventually terminates.
5070:"Convex polytopes, algebraic geometry, and combinatorics"
4786:"An optimal convex hull algorithm in any fixed dimension"
4350:
Convex Sets and Their Applications. Summer Lectures 1959.
3896:
3894:
3892:
3646:, Theorem 1.1.2 (pages 2–3) and Chapter 3.
1941:
of the polygon and a single convex hull edge, are called
5019:(1983), "On the shape of a set of points in the plane",
4194:
215:
Convex hull of a bounded planar set: rubber band analogy
5300:(1983), "Finding the convex hull of a simple polygon",
907:, and the third and fourth definitions are equivalent.
611:. Therefore, the first two definitions are equivalent.
3871:
2895:
of solutions to combinatorial problems are central to
6838:
Whitley, Robert (1986), "The KreÄn-Ĺ mulian theorem",
6786:
Westermann, L. R. J. (1976), "On the hull operator",
6060:
Mainstream Mathematical Economics in the 20th Century
5387:"Mathematical models for statistical decision theory"
4686:(2012), "Three problems about dynamic convex hulls",
4198:
2794:
2765:
2669:
2642:
2622:
2473:
2447:
2401:
2359:
2339:
2315:
2295:
2275:
2206:
2182:
2158:
2059:
2035:
1995:
1975:
1897:
1847:
1827:
1807:
1772:
1752:
1719:
1689:
1650:
1567:
1547:
1527:
1498:
1478:
1452:
1432:
1412:
1383:
1363:
1198:
1144:
1120:
1097:
1077:
1057:
1026:
1002:
982:
958:
893:
873:
833:
813:
787:
767:
747:
727:
700:
680:
660:
640:
620:
597:
577:
557:
537:
517:
497:
477:
410:
390:
366:
331:
305:
279:
257:
233:
128:
6455:
Sakuma, Itsuo (1977), "Closedness of convex hulls",
6152:
Okon, T. (2000), "Choquet theory in metric spaces",
3531:
3217:
based on points where the animal has been observed.
3166:, it can be made convex by taking convex hulls. The
2788:
can be viewed as the projection of a convex hull in
7419:
7386:
7341:
7272:
7198:
7122:
7064:
7038:
6818:White, F. Puryer (April 1923), "Pure mathematics",
4676:
Computational Geometry: Algorithms and Applications
3112:, central objects of study are the convex hulls of
43:The convex hull of the red set is the blue and red
5188:(1994), "6. Newton Polytopes and Chow Polytopes",
4110:
3363:refers to the fact that the convex hull defines a
2818:
2780:
2739:, obtained as an intersection of sublevel sets of
2683:
2648:
2628:
2509:
2459:
2425:
2383:
2345:
2321:
2301:
2281:
2212:
2188:
2164:
2082:
2041:
2021:
1981:
1903:
1883:
1833:
1813:
1778:
1758:
1734:
1701:
1671:
1573:
1553:
1533:
1504:
1484:
1464:
1438:
1418:
1389:
1369:
1267:
1153:
1126:
1106:
1083:
1063:
1032:
1008:
988:
964:
899:
879:
851:
819:
799:
773:
753:
733:
706:
686:
666:
646:
626:
603:
583:
563:
543:
523:
503:
483:
416:
396:
372:
337:
311:
285:
263:
239:
152:
5694:Transactions of the American Mathematical Society
5068:Escobar, Laura; Kaveh, Kiumars (September 2020),
4902:(2008), "All polygons flip finitely ... right?",
4594:Transactions of the American Mathematical Society
3799:
3416:
3371:, with which it should not be confused — see e.g
1221:
6841:Proceedings of the American Mathematical Society
6123:Proceedings of the American Mathematical Society
5222:Getz, Wayne M.; Wilmers, Christopher C. (2004),
5134:Proceedings of the American Mathematical Society
4751:(1985), "On the convex layers of a planar set",
4495:Proceedings of the American Mathematical Society
4455:(1997), "How good are convex hull algorithms?",
3727:
6659:Journal fĂĽr die Reine und Angewandte Mathematik
4137:; see especially remarks following Theorem 2.9.
3751:
2395:are known that compute the convex hull in time
273:The intersection of all convex sets containing
4904:Surveys on Discrete and Computational Geometry
3034:, and applied to determine the equivalence of
887:belongs to a simplex whose vertices belong to
98:. The convex hull operator is an example of a
7012:
6521:: CS1 maint: DOI inactive as of March 2024 (
6154:Zeitschrift fĂĽr Analysis und ihre Anwendungen
5498:(1976), "Normality and the numerical range",
5356:Bulletin of the American Mathematical Society
4086:
3928:Edelsbrunner, Kirkpatrick & Seidel (1983)
3459:
3457:
3421:answer to "the perimeter of a non-convex set"
2851:, but the exponent of the algorithm is high.
2636:such that every ray from a viewpoint through
2262:of facets and their adjacencies, or the full
2172:on a real vector space is the function whose
1742:. Each extreme point of the hull is called a
8:
6433:(1999), "The bagplot: A bivariate boxplot",
5077:Notices of the American Mathematical Society
4492:(1982), "Quantitative Helly-type theorems",
4413:Notices of the American Mathematical Society
4304:
4257:
4233:
3952:Ottmann, Soisalon-Soininen & Wood (1984)
3839:
3639:
2499:
2485:
2467:, the time for computing the convex hull is
2176:is the lower convex hull of the epigraph of
2106:, the convex hull of two circles in 3d space
1873:
1859:
1801:, the number of faces of the convex hull of
1357:, meaning that the convex hull of every set
1278:(the set of points that lie on or above the
4540:(1999), "Data structures for mobile data",
4245:
4221:
4134:
3787:
3775:
3587:
3511:
3463:
3387:
2735:is a generalization of similar concepts to
251:The (unique) minimal convex set containing
7019:
7005:
6997:
5835:"On extreme points of regular convex sets"
5109:, Cambridge University Press, p. 55,
4028:Gel'fand, Kapranov & Zelevinsky (1994)
3912:
3900:
3628:Kashiwabara, Nakamura & Okamoto (2005)
3120:can be used to find optimal solutions. In
2720:, which can be thought of as the smallest
2510:{\displaystyle O(n^{\lfloor d/2\rfloor })}
1884:{\displaystyle O(n^{\lfloor d/2\rfloor })}
27:Smallest convex set containing a given set
6820:Science Progress in the Twentieth Century
6801:
6769:
6730:
6628:
6545:
6483:Convex Bodies: The Brunn–Minkowski Theory
6318:
6165:
6135:
5887:
5852:
5706:
5658:
5626:
5545:
5513:
5478:
5368:
4866:
4806:
4731:
4555:
4507:
4389:Algebraic Numbers and Algebraic Functions
3939:
3883:
3843:
3703:
3643:
3189:In the geometry of boat and ship design,
2801:
2797:
2796:
2793:
2772:
2768:
2767:
2764:
2673:
2668:
2641:
2621:
2491:
2484:
2472:
2446:
2400:
2358:
2338:
2314:
2294:
2274:
2205:
2181:
2157:
2069:
2058:
2034:
2022:{\displaystyle \pi /2<\theta <\pi }
1999:
1994:
1974:
1896:
1865:
1858:
1846:
1826:
1806:
1771:
1751:
1726:
1722:
1721:
1718:
1688:
1672:{\displaystyle S\subset \mathbb {R} ^{d}}
1663:
1659:
1658:
1649:
1566:
1546:
1526:
1497:
1477:
1451:
1431:
1411:
1382:
1362:
1251:
1235:
1220:
1219:
1197:
1143:
1119:
1096:
1076:
1056:
1025:
1001:
981:
957:
892:
872:
832:
812:
786:
766:
746:
726:
699:
679:
659:
639:
619:
596:
576:
556:
536:
516:
496:
476:
409:
389:
365:
330:
304:
278:
256:
232:
127:
6349:Quantum Computing: A Gentle Introduction
5784:(December 1922), "Über konvexe Körper",
4405:"The mathematics of Grace Murray Hopper"
4195:Kernohan, Gitzen & Millspaugh (2001)
4170:
4038:
4011:
3999:
3963:
3823:
3667:
3615:
3542:
3523:
3320:
1541:, the convex hull of the convex hull of
781:is also a convex combination of at most
7109:Locally convex topological vector space
6723:Proceedings of IEEE MELECON '83, Athens
5022:IEEE Transactions on Information Theory
4754:IEEE Transactions on Information Theory
4281:
4062:
4050:
3575:
3436:
3352:
3049:for their application to the theory of
2875:involve convex hulls. They are used in
1326:locally convex topological vector space
6580:Trudy Seminara imeni I. G. Petrovskogo
6514:
4296:
4209:
4158:
4122:
3872:Basch, Guibas & Hershberger (1999)
3763:
3739:
3642:, Theorem 3, pages 562–563;
3563:
3550:
3527:
3499:
3487:
1644:The convex hull of a finite point set
1165:Preservation of topological properties
827:. The set of convex combinations of a
6611:"Remarks on piecewise-linear algebra"
5581:Radio Tracking and Animal Populations
4794:Discrete & Computational Geometry
4719:Discrete & Computational Geometry
4391:, Gordon and Breach, pp. 37–43,
4329:
4317:
4285:
4269:
4199:Nilsen, Pedersen & Linnell (2008)
4182:
4146:
4098:
4023:
3987:
3975:
3448:
3336:
1185:Topologically, the convex hull of an
1020:itself (as happens, for instance, if
7:
6182:Ottmann, T.; Soisalon-Soininen, E.;
5286:, Longmans, Green, & Co., 1906,
4300:
4074:
3859:
3603:
3532:Bárány, Katchalski & Pach (1982)
3226:method by combining convex hulls of
3150:, agents are assumed to have convex
3042:
2224:) and, in this form, is dual to the
1297:(a subset that is compact under the
384:so that it surrounds the entire set
75:, or equivalently as the set of all
32:Hull (watercraft) § Hull shapes
6533:Journal of Mathematics and the Arts
6000:Encyclopaedia of Ships and Shipping
5501:Linear Algebra and Its Applications
3847:
3591:
3546:
3467:
3372:
3197:of the vessel. It differs from the
2819:{\displaystyle \mathbb {R} ^{n+1}.}
1406:, meaning that, for every two sets
6700:KĹŤdai Mathematical Seminar Reports
5934:Convex Sets and their Applications
4111:Rousseeuw, Ruts & Tukey (1999)
2289:, the number of input points, and
1634:Convex hull of points in the plane
1561:is the same as the convex hull of
1492:is a subset of the convex hull of
1336:Geometric and algebraic properties
972:is the intersection of all closed
25:
6746:, North-Holland, pp. 853–856
4990:Journal for Geometry and Graphics
4509:10.1090/S0002-9939-1982-0663877-X
3800:Avis, Bremner & Seidel (1997)
3327:appears earlier, for instance in
3323:), and the corresponding term in
3182:, one of the key properties of a
3046:
2724:containing the points of a given
2698:of a subset of a two-dimensional
1967:continuously differentiable curve
1051:If the open convex hull of a set
467:3D convex hull of 120 point cloud
5253:10.1111/j.0906-7590.2004.03835.x
3728:Cranston, Hsu & March (1989)
2781:{\displaystyle \mathbb {R} ^{n}}
2570:
2556:
1841:-dimensional Euclidean space is
1735:{\displaystyle \mathbb {R} ^{d}}
7214:Ekeland's variational principle
6352:, MIT Press, pp. 215–216,
6210:"1.2.1 The Gauss–Lucas theorem"
5370:10.1090/S0002-9904-1947-08787-5
4833:Computer Aided Geometric Design
3417:Williams & Rossignac (2005)
3248:positive-semidefinite operators
2333:can compute the convex hull of
2132:Alexandrov's uniqueness theorem
2083:{\displaystyle 1-\pi /2\theta }
1921:Convex hull of a simple polygon
1138:provide examples where exactly
380:. One may imagine stretching a
79:of points in the subset. For a
6991:Wolfram Demonstrations Project
6616:Pacific Journal of Mathematics
6449:10.1080/00031305.1999.10474494
6253:, Springer, pp. 312–345,
6062:, Springer, pp. 197–215,
6015:Information Processing Letters
5646:Foundations of Physics Letters
5106:Phase Transitions in Materials
4983:"The development of the oloid"
4632:Information Processing Letters
4361:Information Processing Letters
2980:describes the convex hulls of
2706:between any two of its points.
2504:
2477:
2420:
2405:
2378:
2363:
1878:
1851:
1216:
1204:
846:
834:
147:
132:
118:, are fundamental problems of
90:are open, and convex hulls of
1:
6757:Topology and Its Applications
6593:Journal of Soviet Mathematics
6556:10.1080/17513472.2017.1318512
6346:; Polak, Wolfgang H. (2011),
6306:Israel Journal of Mathematics
6137:10.1090/S0002-9939-07-08887-9
5708:10.1090/S0002-9947-02-02915-X
5466:Topology and Its Applications
4942:American Mathematical Monthly
4845:10.1016/s0167-8396(03)00003-7
4471:10.1016/S0925-7721(96)00023-5
3752:Dirnböck & Stachel (1997)
122:. They can be solved in time
6803:10.1016/1385-7258(76)90065-2
6771:10.1016/0166-8641(93)90032-9
6471:10.1016/0022-0531(77)90095-3
6259:10.1007/978-3-642-68874-4_13
6208:Prasolov, Victor V. (2004),
6202:10.1016/0020-0255(84)90025-2
6068:10.1007/978-3-662-04238-0_16
6028:10.1016/0020-0190(79)90069-3
5556:10.1016/j.comgeo.2004.05.001
5515:10.1016/0024-3795(76)90080-x
5480:10.1016/0166-8641(92)90092-E
5316:10.1016/0196-6774(83)90013-5
4645:10.1016/0020-0190(79)90074-7
4374:10.1016/0020-0190(79)90072-3
3148:general economic equilibrium
3122:multi-objective optimization
2879:as the outermost contour of
2439:Kirkpatrick–Seidel algorithm
2353:points in the plane in time
1465:{\displaystyle X\subseteq Y}
937:of the convex hull, and the
7234:Hermite–Hadamard inequality
6952:Encyclopedia of Mathematics
6047:"Letter to Henry Oldenburg"
5103:Fultz, Brent (April 2020),
4353:, Argon national laboratory
4087:Epstein & Marden (1987)
2733:holomorphically convex hull
2393:output-sensitive algorithms
2196:. It is the unique maximal
614:Each convex set containing
67:of a shape is the smallest
7484:
6458:Journal of Economic Theory
5997:Mason, Herbert B. (1908),
5619:10.1038/s41598-019-56497-6
4305:Escobar & Kaveh (2020)
4234:Rieffel & Polak (2011)
3840:McCallum & Avis (1979)
3640:Krein & Ĺ mulian (1940)
3135:
3106:combinatorial optimization
3100:Combinatorial optimization
2972:is the convex hull of its
2897:combinatorial optimization
2871:, and several theorems in
2737:complex analytic manifolds
2426:{\displaystyle O(n\log h)}
2384:{\displaystyle O(n\log n)}
2235:
2141:
1918:
1637:
1308:
952:The closed convex hull of
459:Equivalence of definitions
153:{\displaystyle O(n\log n)}
29:
6880:, ACM, pp. 107–112,
6789:Indagationes Mathematicae
6694:Talman, Louis A. (1977),
6595:33 (4): 1140–1153, 1986,
6436:The American Statistician
6289:10.1111/1467-8659.1140235
6222:10.1007/978-3-642-03980-5
6106:10.1007/s11284-007-0421-9
5936:, John Wiley & Sons,
5787:Mathematische Zeitschrift
5677:10.1007/s10702-006-1852-1
5198:10.1007/978-0-8176-4771-1
4702:10.1142/S0218195912600096
4222:Getz & Wilmers (2004)
3588:Krein & Milman (1940)
3562:This example is given by
3419:. See also Douglas Zare,
3017:geometrization conjecture
2953:, according to which the
2889:randomized decision rules
2684:{\displaystyle 1/\alpha }
2536:method for computing the
2110:For the convex hull of a
1969:. However, for any angle
1766:and that encloses all of
1521:, meaning that for every
7420:Applications and related
7224:Fenchel-Young inequality
6492:10.1017/CBO9780511526282
5385:Harris, Bernard (1971),
5035:10.1109/TIT.1983.1056714
4939:(1938), "On convexity",
4767:10.1109/TIT.1985.1057060
4678:(3rd ed.), Springer
3110:polyhedral combinatorics
3094:randomized decision rule
3041:See also the section on
3021:low-dimensional topology
2901:polyhedral combinatorics
2691:that contain the subset.
1786:. For sets of points in
996:. If the convex hull of
945:(or in some sources the
114:problem of intersecting
7180:Legendre transformation
7104:Legendre transformation
6886:10.1145/1060244.1060257
6672:10.1515/crll.1914.144.1
6630:10.2140/pjm.1982.98.183
6494:(inactive 2024-03-18),
6367:Rockafellar, R. Tyrrell
6277:Computer Graphics Forum
5932:Lay, Steven R. (1982),
4385:"2.5. Newton's Polygon"
3844:Graham & Yao (1983)
3252:Schrödinger–HJW theorem
3168:Shapley–Folkman theorem
3158:. These assumptions of
3126:quasiconvex combination
2751:of a point set and its
2743:containing a given set.
2122:. Examples include the
2042:{\displaystyle \theta }
1982:{\displaystyle \theta }
1961:The curve generated by
1947:convex differences tree
1790:, the convex hull is a
1603:Shapley–Folkman theorem
863:; in the plane it is a
7463:Computational geometry
7427:Convexity in economics
7361:(lower) ideally convex
7219:Fenchel–Moreau theorem
7209:Carathéodory's theorem
6053:, University of Oxford
5854:10.4064/sm-9-1-133-138
5533:Computational Geometry
4912:10.1090/conm/453/08801
4900:Toussaint, Godfried T.
4868:10.1214/aop/1176991500
4566:10.1006/jagm.1998.0988
4458:Computational Geometry
4171:Chen & Wang (2003)
4012:Chang & Yap (1986)
3283:
3160:convexity in economics
3138:Convexity in economics
3070:
2931:
2905:convexity in economics
2820:
2782:
2749:Delaunay triangulation
2722:injective metric space
2711:orthogonal convex hull
2685:
2650:
2630:
2564:Orthogonal convex hull
2511:
2461:
2460:{\displaystyle d>3}
2427:
2385:
2347:
2323:
2303:
2283:
2244:computational geometry
2238:Convex hull algorithms
2214:
2190:
2166:
2107:
2084:
2043:
2023:
1983:
1930:
1905:
1885:
1835:
1815:
1780:
1760:
1736:
1709:, or more generally a
1703:
1673:
1635:
1575:
1555:
1535:
1506:
1486:
1466:
1440:
1420:
1391:
1371:
1269:
1182:
1155:
1128:
1108:
1085:
1065:
1034:
1010:
990:
966:
949:) of the convex hull.
920:Topological properties
901:
881:
859:-tuple of points is a
853:
821:
801:
775:
755:
735:
719:Carathéodory's theorem
717:In fact, according to
708:
688:
668:
648:
628:
605:
585:
565:
545:
525:
505:
485:
468:
441:non-Euclidean geometry
418:
398:
374:
339:
313:
287:
265:
241:
216:
193:Delaunay triangulation
185:orthogonal convex hull
181:epigraphs of functions
164:in higher dimensions.
154:
120:computational geometry
48:
7349:Convex series related
7249:Shapley–Folkman lemma
6391:Annals of Mathematics
6313:(3): 238–244 (1980),
5867:Annals of Mathematics
5811:; see also review by
5738:10.1007/3-540-55611-7
5303:Journal of Algorithms
5239:(4), Wiley: 489–505,
5115:10.1017/9781108641449
5013:Kirkpatrick, David G.
5009:Edelsbrunner, Herbert
4854:Annals of Probability
4543:Journal of Algorithms
3828:de Berg et al. (2008)
3812:de Berg et al. (2008)
3716:Demaine et al. (2008)
3692:de Berg et al. (2008)
3680:de Berg et al. (2008)
3656:de Berg et al. (2008)
3480:de Berg et al. (2008)
3405:de Berg et al. (2008)
3265:
3064:
2926:
2821:
2783:
2741:holomorphic functions
2686:
2651:
2631:
2512:
2462:
2428:
2386:
2348:
2324:
2304:
2284:
2215:
2191:
2167:
2150:lower convex envelope
2144:Lower convex envelope
2101:
2085:
2044:
2024:
1984:
1933:The convex hull of a
1928:
1906:
1886:
1836:
1816:
1781:
1761:
1737:
1704:
1674:
1633:
1576:
1556:
1536:
1507:
1487:
1472:, the convex hull of
1467:
1441:
1421:
1392:
1372:
1301:) is weakly compact.
1291:Krein–Smulian theorem
1270:
1172:
1156:
1129:
1109:
1086:
1066:
1035:
1011:
991:
967:
925:Closed and open hulls
911:Upper and lower hulls
902:
882:
854:
852:{\displaystyle (d+1)}
822:
802:
776:
756:
736:
709:
689:
669:
649:
629:
606:
586:
566:
546:
526:
506:
486:
466:
419:
399:
375:
340:
314:
288:
266:
242:
219:A set of points in a
214:
155:
42:
7239:Krein–Milman theorem
7032:variational analysis
6189:Information Sciences
6045:(October 24, 1676),
5432:10.1007/128_2013_486
5396:, pp. 369–389,
4488:; Katchalski, Meir;
4403:Auel, Asher (2019),
3317:Garrett Birkhoff
3292:Josiah Willard Gibbs
3092:, the risk set of a
3051:developable surfaces
3032:hyperbolic manifolds
2792:
2763:
2696:relative convex hull
2667:
2640:
2620:
2578:Relative convex hull
2471:
2445:
2399:
2357:
2337:
2313:
2293:
2273:
2204:
2180:
2156:
2057:
2033:
1993:
1973:
1895:
1845:
1825:
1805:
1770:
1750:
1717:
1687:
1648:
1565:
1545:
1525:
1496:
1476:
1450:
1430:
1410:
1381:
1361:
1322:Krein–Milman theorem
1311:Krein–Milman theorem
1286:as its convex hull.
1196:
1142:
1118:
1095:
1075:
1055:
1044:or more generally a
1024:
1000:
980:
956:
891:
871:
831:
811:
785:
765:
745:
725:
698:
678:
658:
638:
618:
595:
575:
555:
535:
515:
495:
475:
408:
388:
364:
329:
303:
277:
255:
231:
126:
7468:Geometry processing
7229:Jensen's inequality
7099:Lagrange multiplier
7089:Convex optimization
7084:Convex metric space
6740:Toussaint, Godfried
6719:Toussaint, Godfried
6427:Rousseeuw, Peter J.
6344:Rieffel, Eleanor G.
6098:2008EcoR...23..635N
6086:Ecological Research
5669:2006FoPhL..19...95K
5611:2019NatSR...920253K
5496:Johnson, Charles R.
5245:2004Ecogr..27..489G
4894:; Gassend, Blaise;
4534:Guibas, Leonidas J.
2951:Gauss–Lucas theorem
2529:kinetic convex hull
2525:Dynamic convex hull
2252:linear inequalities
2148:The convex hull or
2051:Hausdorff dimension
1799:upper bound theorem
1792:simplicial polytope
1702:{\displaystyle d=2}
1161:points are needed.
800:{\displaystyle d+1}
354:simple closed curve
297:convex combinations
162:upper bound theorem
77:convex combinations
7357:(cs, bcs)-complete
7328:Algebraic interior
7046:Convex combination
6966:Weisstein, Eric W.
6607:Sontag, Eduardo D.
6601:10.1007/BF01086114
6320:10.1007/BF02760885
6247:Pulleyblank, W. R.
6051:The Newton Project
6008:McCallum, Duncan;
5975:10.1007/BF00993195
5889:10338.dmlcz/100106
5840:Studia Mathematica
5800:10.1007/bf01215899
5599:Scientific Reports
5583:, Academic Press,
4817:10.1007/BF02573985
4733:10.1007/BF02187692
4451:; Bremner, David;
4246:Kirkpatrick (2006)
4135:Pulleyblank (1983)
3788:Rockafellar (1970)
3776:Rockafellar (1970)
3512:Rockafellar (1970)
3464:Rockafellar (1970)
3388:Rockafellar (1970)
3369:closed convex hull
3290:was identified by
3284:
3180:geometric modeling
3174:Geometric modeling
3156:convex preferences
3144:Arrow–Debreu model
3118:linear programming
3071:
2998:Tverberg's theorem
2932:
2929:Tverberg's theorem
2891:. Convex hulls of
2883:, are part of the
2816:
2778:
2681:
2646:
2626:
2548:Related structures
2507:
2457:
2423:
2381:
2343:
2319:
2299:
2279:
2210:
2186:
2162:
2108:
2080:
2039:
2019:
1979:
1951:Erdős–Nagy theorem
1931:
1901:
1881:
1831:
1811:
1776:
1756:
1732:
1699:
1669:
1636:
1609:Projective duality
1571:
1551:
1531:
1502:
1482:
1462:
1436:
1416:
1387:
1367:
1265:
1183:
1154:{\displaystyle 2d}
1151:
1124:
1107:{\displaystyle 2d}
1104:
1081:
1061:
1030:
1006:
986:
962:
931:closed convex hull
897:
877:
849:
817:
797:
771:
751:
731:
704:
684:
664:
644:
624:
601:
581:
561:
541:
521:
501:
481:
469:
445:real vector spaces
414:
394:
370:
335:
309:
283:
261:
247:may be defined as
237:
217:
150:
49:
7453:Closure operators
7435:
7434:
6987:Eric W. Weisstein
6752:Weeks, Jeffrey R.
6394:, Second Series,
6359:978-0-262-01506-6
6268:978-3-642-68876-8
6077:978-3-642-08638-0
5926:10.1109/34.273735
5870:, Second Series,
5441:978-3-319-05773-6
5294:Graham, Ronald L.
5268:Gibbs, Willard J.
5186:Zelevinsky, A. V.
5050:Epstein, D. B. A.
5042:Epstein, D. B. A.
4979:Stachel, Hellmuth
4921:978-0-8218-4239-3
4782:Chazelle, Bernard
4749:Chazelle, Bernard
4589:Birkhoff, Garrett
4538:Hershberger, John
4258:Kim et al. (2019)
3913:Laurentini (1994)
3901:Westermann (1976)
3286:A convex hull in
3224:local convex hull
3114:indicator vectors
3075:robust statistics
3005:hyperbolic spaces
2990:discrete geometry
2978:Russo–Dye theorem
2962:spectral analysis
2893:indicator vectors
2877:robust statistics
2873:discrete geometry
2658:3D reconstruction
2649:{\displaystyle p}
2629:{\displaystyle p}
2534:rotating calipers
2441:. For dimensions
2346:{\displaystyle n}
2322:{\displaystyle n}
2302:{\displaystyle h}
2282:{\displaystyle n}
2222:pointwise minimum
2213:{\displaystyle f}
2189:{\displaystyle f}
2165:{\displaystyle f}
1904:{\displaystyle n}
1834:{\displaystyle d}
1814:{\displaystyle n}
1797:According to the
1779:{\displaystyle S}
1759:{\displaystyle S}
1626:Finite point sets
1574:{\displaystyle X}
1554:{\displaystyle X}
1534:{\displaystyle X}
1505:{\displaystyle Y}
1485:{\displaystyle X}
1439:{\displaystyle Y}
1419:{\displaystyle X}
1390:{\displaystyle X}
1377:is a superset of
1370:{\displaystyle X}
1258:
1127:{\displaystyle X}
1084:{\displaystyle d}
1064:{\displaystyle X}
1033:{\displaystyle X}
1009:{\displaystyle X}
989:{\displaystyle X}
965:{\displaystyle X}
947:relative interior
900:{\displaystyle X}
880:{\displaystyle X}
820:{\displaystyle X}
774:{\displaystyle X}
754:{\displaystyle d}
741:is a subset of a
734:{\displaystyle X}
707:{\displaystyle X}
687:{\displaystyle X}
667:{\displaystyle X}
647:{\displaystyle X}
627:{\displaystyle X}
604:{\displaystyle X}
584:{\displaystyle Y}
564:{\displaystyle X}
544:{\displaystyle Y}
524:{\displaystyle X}
504:{\displaystyle X}
484:{\displaystyle X}
453:oriented matroids
417:{\displaystyle S}
397:{\displaystyle S}
373:{\displaystyle X}
338:{\displaystyle X}
325:with vertices in
321:The union of all
312:{\displaystyle X}
286:{\displaystyle X}
264:{\displaystyle X}
240:{\displaystyle X}
223:is defined to be
16:(Redirected from
7475:
7353:(cs, lcs)-closed
7299:Effective domain
7254:Robinson–Ursescu
7130:Convex conjugate
7021:
7014:
7007:
6998:
6979:
6978:
6960:
6934:
6918:(4): 1206–1215,
6904:
6872:
6834:
6814:
6805:
6782:
6773:
6747:
6735:
6734:
6714:
6690:
6649:
6632:
6591:, translated in
6590:
6574:
6549:
6526:
6520:
6512:
6473:
6451:
6422:
6381:
6362:
6339:
6322:
6299:
6271:
6242:
6204:
6178:
6169:
6148:
6139:
6130:(6): 1689–1694,
6116:
6080:
6054:
6038:
6004:
5993:
5954:
5928:
5908:
5891:
5857:
5856:
5810:
5777:
5776:
5775:
5766:, archived from
5729:Axioms and Hulls
5724:Knuth, Donald E.
5719:
5710:
5701:(5): 2035–2053,
5687:
5662:
5660:quant-ph/0305068
5639:
5630:
5593:
5575:
5574:, E75-A: 321–329
5566:
5549:
5526:
5517:
5491:
5482:
5473:(1–3): 181–187,
5452:
5418:
5417:
5416:
5410:
5404:, archived from
5391:
5381:
5372:
5349:
5336:Convex Polytopes
5331:GrĂĽnbaum, Branko
5326:
5279:
5263:
5228:
5218:
5173:
5127:
5099:
5089:10.1090/noti2137
5083:(8): 1116–1123,
5074:
5064:
5037:
5004:
4987:
4977:Dirnböck, Hans;
4973:
4932:
4896:O'Rourke, Joseph
4892:Demaine, Erik D.
4887:
4870:
4847:
4827:
4810:
4790:
4777:
4744:
4735:
4712:
4684:Chan, Timothy M.
4679:
4655:
4625:
4584:
4559:
4528:
4511:
4481:
4465:(5–6): 265–301,
4444:
4426:10.1090/noti1810
4409:
4399:
4376:
4354:
4347:Fan, Ky (1959),
4333:
4327:
4321:
4314:
4308:
4303:, page 336, and
4294:
4288:
4279:
4273:
4267:
4261:
4255:
4249:
4243:
4237:
4231:
4225:
4219:
4213:
4207:
4201:
4192:
4186:
4180:
4174:
4168:
4162:
4156:
4150:
4144:
4138:
4132:
4126:
4120:
4114:
4108:
4102:
4096:
4090:
4084:
4078:
4072:
4066:
4060:
4054:
4048:
4042:
4036:
4030:
4021:
4015:
4009:
4003:
3997:
3991:
3985:
3979:
3973:
3967:
3961:
3955:
3949:
3943:
3940:Toussaint (1986)
3937:
3931:
3925:
3916:
3910:
3904:
3898:
3887:
3884:Toussaint (1983)
3881:
3875:
3869:
3863:
3857:
3851:
3837:
3831:
3821:
3815:
3809:
3803:
3797:
3791:
3785:
3779:
3773:
3767:
3761:
3755:
3749:
3743:
3737:
3731:
3725:
3719:
3713:
3707:
3704:Rappoport (1992)
3701:
3695:
3689:
3683:
3677:
3671:
3665:
3659:
3653:
3647:
3644:Schneider (1993)
3637:
3631:
3625:
3619:
3613:
3607:
3601:
3595:
3585:
3579:
3573:
3567:
3560:
3554:
3540:
3534:
3521:
3515:
3509:
3503:
3497:
3491:
3477:
3471:
3461:
3452:
3446:
3440:
3434:
3428:
3414:
3408:
3402:
3391:
3385:
3376:
3365:closure operator
3359:The terminology
3357:
2982:unitary elements
2943:Newton polytopes
2869:unitary elements
2825:
2823:
2822:
2817:
2812:
2811:
2800:
2787:
2785:
2784:
2779:
2777:
2776:
2771:
2718:hyperconvex hull
2690:
2688:
2687:
2682:
2677:
2655:
2653:
2652:
2647:
2635:
2633:
2632:
2627:
2574:
2560:
2544:of a point set.
2516:
2514:
2513:
2508:
2503:
2502:
2495:
2466:
2464:
2463:
2458:
2435:Chan's algorithm
2433:. These include
2432:
2430:
2429:
2424:
2390:
2388:
2387:
2382:
2352:
2350:
2349:
2344:
2328:
2326:
2325:
2320:
2308:
2306:
2305:
2300:
2288:
2286:
2285:
2280:
2260:undirected graph
2258:of the hull, an
2226:convex conjugate
2219:
2217:
2216:
2211:
2195:
2193:
2192:
2187:
2171:
2169:
2168:
2163:
2089:
2087:
2086:
2081:
2073:
2048:
2046:
2045:
2040:
2028:
2026:
2025:
2020:
2003:
1988:
1986:
1985:
1980:
1910:
1908:
1907:
1902:
1890:
1888:
1887:
1882:
1877:
1876:
1869:
1840:
1838:
1837:
1832:
1820:
1818:
1817:
1812:
1788:general position
1785:
1783:
1782:
1777:
1765:
1763:
1762:
1757:
1741:
1739:
1738:
1733:
1731:
1730:
1725:
1708:
1706:
1705:
1700:
1678:
1676:
1675:
1670:
1668:
1667:
1662:
1580:
1578:
1577:
1572:
1560:
1558:
1557:
1552:
1540:
1538:
1537:
1532:
1511:
1509:
1508:
1503:
1491:
1489:
1488:
1483:
1471:
1469:
1468:
1463:
1445:
1443:
1442:
1437:
1425:
1423:
1422:
1417:
1396:
1394:
1393:
1388:
1376:
1374:
1373:
1368:
1347:closure operator
1341:Closure operator
1284:upper half-plane
1274:
1272:
1271:
1266:
1264:
1260:
1259:
1257:
1256:
1255:
1236:
1225:
1224:
1179:upper half-plane
1160:
1158:
1157:
1152:
1133:
1131:
1130:
1125:
1113:
1111:
1110:
1105:
1090:
1088:
1087:
1082:
1070:
1068:
1067:
1062:
1039:
1037:
1036:
1031:
1015:
1013:
1012:
1007:
995:
993:
992:
987:
971:
969:
968:
963:
939:open convex hull
933:of a set is the
906:
904:
903:
898:
886:
884:
883:
878:
858:
856:
855:
850:
826:
824:
823:
818:
806:
804:
803:
798:
780:
778:
777:
772:
760:
758:
757:
752:
740:
738:
737:
732:
713:
711:
710:
705:
693:
691:
690:
685:
673:
671:
670:
665:
653:
651:
650:
645:
633:
631:
630:
625:
610:
608:
607:
602:
590:
588:
587:
582:
570:
568:
567:
562:
550:
548:
547:
542:
530:
528:
527:
522:
510:
508:
507:
502:
490:
488:
487:
482:
430:obstacle problem
423:
421:
420:
415:
403:
401:
400:
395:
379:
377:
376:
371:
344:
342:
341:
336:
318:
316:
315:
310:
292:
290:
289:
284:
270:
268:
267:
262:
246:
244:
243:
238:
159:
157:
156:
151:
100:closure operator
86:Convex hulls of
21:
7483:
7482:
7478:
7477:
7476:
7474:
7473:
7472:
7458:Convex analysis
7438:
7437:
7436:
7431:
7415:
7382:
7337:
7268:
7194:
7185:Semi-continuity
7170:Convex function
7151:Logarithmically
7118:
7079:Convex geometry
7060:
7051:Convex function
7034:
7028:Convex analysis
7025:
6964:
6963:
6945:
6942:
6937:
6924:10.2307/2533254
6907:
6875:
6854:10.2307/2046536
6837:
6826:(68): 517–526,
6817:
6785:
6750:
6738:
6732:10.1.1.155.5671
6717:
6693:
6652:
6605:
6577:
6529:
6513:
6502:
6478:Schneider, Rolf
6476:
6454:
6425:
6404:10.2307/1970292
6384:
6371:Convex Analysis
6365:
6360:
6342:
6302:
6274:
6269:
6245:
6232:
6207:
6181:
6167:10.4171/ZAA/952
6151:
6119:
6083:
6078:
6057:
6041:
6007:
5996:
5957:
5944:
5931:
5911:
5880:10.2307/1968735
5860:
5825:
5813:Hans Rademacher
5780:
5773:
5771:
5748:
5722:
5690:
5642:
5596:
5591:
5578:
5569:
5529:
5494:
5461:Herrlich, Horst
5459:
5442:
5421:
5414:
5412:
5408:
5389:
5384:
5352:
5347:
5329:
5298:Yao, F. Frances
5292:
5280:; reprinted in
5266:
5226:
5221:
5208:
5182:Kapranov, M. M.
5178:Gel'fand, I. M.
5176:
5147:10.2307/2044692
5130:
5125:
5102:
5072:
5067:
5040:
5017:Seidel, Raimund
5007:
4985:
4976:
4955:10.2307/2302604
4935:
4922:
4890:
4850:
4830:
4808:10.1.1.113.8709
4788:
4780:
4747:
4715:
4682:
4672:Schwarzkopf, O.
4664:van Kreveld, M.
4658:
4628:
4607:10.2307/1989687
4587:
4557:10.1.1.134.6921
4532:Basch, Julien;
4531:
4484:
4453:Seidel, Raimund
4447:
4407:
4402:
4379:
4357:
4346:
4342:
4337:
4336:
4328:
4324:
4315:
4311:
4295:
4291:
4280:
4276:
4268:
4264:
4256:
4252:
4244:
4240:
4232:
4228:
4220:
4216:
4208:
4204:
4193:
4189:
4181:
4177:
4169:
4165:
4157:
4153:
4145:
4141:
4133:
4129:
4121:
4117:
4109:
4105:
4097:
4093:
4085:
4081:
4073:
4069:
4061:
4057:
4049:
4045:
4039:Prasolov (2004)
4037:
4033:
4022:
4018:
4010:
4006:
4000:Chazelle (1985)
3998:
3994:
3986:
3982:
3974:
3970:
3964:Herrlich (1992)
3962:
3958:
3950:
3946:
3938:
3934:
3926:
3919:
3911:
3907:
3899:
3890:
3882:
3878:
3870:
3866:
3858:
3854:
3838:
3834:
3824:Chazelle (1993)
3822:
3818:
3810:
3806:
3798:
3794:
3786:
3782:
3774:
3770:
3762:
3758:
3750:
3746:
3738:
3734:
3726:
3722:
3714:
3710:
3702:
3698:
3690:
3686:
3678:
3674:
3668:GrĂĽnbaum (2003)
3666:
3662:
3654:
3650:
3638:
3634:
3626:
3622:
3616:Kiselman (2002)
3614:
3610:
3602:
3598:
3586:
3582:
3574:
3570:
3561:
3557:
3543:GrĂĽnbaum (2003)
3541:
3537:
3524:Steinitz (1914)
3522:
3518:
3510:
3506:
3498:
3494:
3478:
3474:
3462:
3455:
3447:
3443:
3435:
3431:
3427:, May 16, 2014.
3415:
3411:
3403:
3394:
3386:
3379:
3358:
3354:
3349:
3329:Hans Rademacher
3313:Henry Oldenburg
3305:
3296:stoichiometries
3281:
3277:
3266:Convex hull of
3260:
3240:quantum physics
3236:
3234:Quantum physics
3207:
3176:
3140:
3134:
3102:
3090:decision theory
3088:In statistical
3059:
3043:Brownian motion
2994:Radon's theorem
2966:numerical range
2935:Newton polygons
2921:
2857:
2849:polynomial time
2795:
2790:
2789:
2766:
2761:
2760:
2757:Voronoi diagram
2665:
2664:
2638:
2637:
2618:
2617:
2584:
2583:
2582:
2581:
2580:
2575:
2567:
2566:
2561:
2550:
2480:
2469:
2468:
2443:
2442:
2397:
2396:
2355:
2354:
2335:
2334:
2311:
2310:
2291:
2290:
2271:
2270:
2254:describing the
2240:
2234:
2202:
2201:
2198:convex function
2178:
2177:
2154:
2153:
2146:
2140:
2096:
2055:
2054:
2031:
2030:
1991:
1990:
1971:
1970:
1963:Brownian motion
1959:
1957:Brownian motion
1939:polygonal chain
1923:
1917:
1915:Simple polygons
1893:
1892:
1854:
1843:
1842:
1823:
1822:
1803:
1802:
1768:
1767:
1748:
1747:
1720:
1715:
1714:
1711:convex polytope
1685:
1684:
1657:
1646:
1645:
1642:
1640:Convex polytope
1628:
1623:
1615:projective dual
1611:
1595:
1563:
1562:
1543:
1542:
1523:
1522:
1494:
1493:
1474:
1473:
1448:
1447:
1428:
1427:
1408:
1407:
1379:
1378:
1359:
1358:
1343:
1338:
1313:
1307:
1282:) has the open
1280:witch of Agnesi
1247:
1240:
1203:
1199:
1194:
1193:
1175:witch of Agnesi
1167:
1140:
1139:
1116:
1115:
1093:
1092:
1073:
1072:
1053:
1052:
1022:
1021:
998:
997:
978:
977:
954:
953:
927:
922:
913:
889:
888:
869:
868:
829:
828:
809:
808:
783:
782:
763:
762:
743:
742:
723:
722:
696:
695:
676:
675:
656:
655:
636:
635:
616:
615:
593:
592:
573:
572:
553:
552:
533:
532:
513:
512:
493:
492:
473:
472:
461:
437:bounding volume
406:
405:
386:
385:
362:
361:
327:
326:
301:
300:
295:The set of all
275:
274:
253:
252:
229:
228:
221:Euclidean space
209:
197:Voronoi diagram
173:Brownian motion
169:simple polygons
124:
123:
73:Euclidean space
61:convex envelope
35:
28:
23:
22:
15:
12:
11:
5:
7481:
7479:
7471:
7470:
7465:
7460:
7455:
7450:
7440:
7439:
7433:
7432:
7430:
7429:
7423:
7421:
7417:
7416:
7414:
7413:
7408:
7406:Strong duality
7403:
7398:
7392:
7390:
7384:
7383:
7381:
7380:
7345:
7343:
7339:
7338:
7336:
7335:
7330:
7321:
7316:
7314:John ellipsoid
7311:
7306:
7301:
7296:
7282:
7276:
7274:
7270:
7269:
7267:
7266:
7261:
7256:
7251:
7246:
7241:
7236:
7231:
7226:
7221:
7216:
7211:
7205:
7203:
7201:results (list)
7196:
7195:
7193:
7192:
7187:
7182:
7177:
7175:Invex function
7172:
7163:
7158:
7153:
7148:
7143:
7137:
7132:
7126:
7124:
7120:
7119:
7117:
7116:
7111:
7106:
7101:
7096:
7091:
7086:
7081:
7076:
7074:Choquet theory
7070:
7068:
7062:
7061:
7059:
7058:
7053:
7048:
7042:
7040:
7039:Basic concepts
7036:
7035:
7026:
7024:
7023:
7016:
7009:
7001:
6995:
6994:
6980:
6961:
6941:
6940:External links
6938:
6936:
6935:
6905:
6873:
6848:(2): 376–377,
6835:
6815:
6796:(2): 179–184,
6783:
6764:(2): 127–149,
6748:
6736:
6715:
6706:(1–2): 62–70,
6691:
6650:
6623:(1): 183–201,
6603:
6582:(6): 239–256,
6575:
6540:(4): 187–202,
6527:
6500:
6474:
6465:(1): 223–227,
6452:
6443:(4): 382–387,
6431:Tukey, John W.
6423:
6398:(3): 470–493,
6382:
6363:
6358:
6340:
6300:
6283:(4): 235–240,
6272:
6267:
6243:
6230:
6205:
6196:(3): 157–171,
6179:
6160:(2): 303–314,
6149:
6117:
6092:(3): 635–639,
6081:
6076:
6055:
6039:
6022:(5): 201–206,
6005:
5994:
5955:
5942:
5929:
5920:(2): 150–162,
5909:
5874:(3): 556–583,
5858:
5823:
5794:(1): 208–210,
5778:
5746:
5720:
5688:
5640:
5594:
5589:
5576:
5567:
5547:10.1.1.14.4965
5540:(2): 129–144,
5527:
5492:
5457:
5440:
5419:
5382:
5363:(4): 299–301,
5350:
5345:
5327:
5310:(4): 324–331,
5290:
5264:
5219:
5206:
5174:
5128:
5123:
5100:
5065:
5038:
5029:(4): 551–559,
5005:
4996:(2): 105–118,
4974:
4949:(4): 199–209,
4933:
4920:
4888:
4861:(1): 144–150,
4848:
4828:
4801:(1): 377–409,
4778:
4761:(4): 509–517,
4745:
4726:(2): 155–182,
4713:
4696:(4): 341–364,
4680:
4668:Overmars, Mark
4656:
4639:(5): 223–228,
4626:
4601:(2): 357–378,
4585:
4529:
4502:(1): 109–114,
4482:
4445:
4420:(3): 330–340,
4400:
4377:
4368:(5): 216–219,
4355:
4343:
4341:
4338:
4335:
4334:
4322:
4309:
4289:
4282:Hautier (2014)
4274:
4262:
4250:
4238:
4226:
4214:
4202:
4197:, p. 137–140;
4187:
4175:
4163:
4151:
4139:
4127:
4115:
4103:
4091:
4079:
4067:
4063:Gardner (1984)
4055:
4051:Johnson (1976)
4043:
4031:
4016:
4004:
3992:
3980:
3968:
3956:
3944:
3932:
3917:
3905:
3888:
3876:
3864:
3852:
3832:
3816:
3804:
3792:
3790:, p. 149.
3780:
3768:
3756:
3744:
3732:
3720:
3708:
3696:
3694:, p. 245.
3684:
3682:, p. 256.
3672:
3660:
3658:, p. 254.
3648:
3632:
3620:
3608:
3596:
3580:
3576:Whitley (1986)
3568:
3555:
3535:
3516:
3504:
3492:
3472:
3453:
3441:
3437:Oberman (2007)
3429:
3409:
3392:
3377:
3361:convex closure
3351:
3350:
3348:
3345:
3304:
3301:
3288:thermodynamics
3279:
3275:
3259:
3258:Thermodynamics
3256:
3235:
3232:
3206:
3203:
3175:
3172:
3136:Main article:
3133:
3130:
3101:
3098:
3058:
3055:
3028:triangulations
3013:ruled surfaces
2937:of univariate
2920:
2917:
2856:
2853:
2815:
2810:
2807:
2804:
2799:
2775:
2770:
2745:
2744:
2729:
2714:
2707:
2700:simple polygon
2692:
2680:
2676:
2672:
2661:
2645:
2625:
2610:
2603:
2596:
2587:For instance:
2576:
2569:
2568:
2562:
2555:
2554:
2553:
2552:
2551:
2549:
2546:
2506:
2501:
2498:
2494:
2490:
2487:
2483:
2479:
2476:
2456:
2453:
2450:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2342:
2318:
2298:
2278:
2248:representation
2236:Main article:
2233:
2230:
2209:
2185:
2161:
2152:of a function
2142:Main article:
2139:
2136:
2120:ruled surfaces
2095:
2092:
2079:
2076:
2072:
2068:
2065:
2062:
2038:
2018:
2015:
2012:
2009:
2006:
2002:
1998:
1978:
1958:
1955:
1935:simple polygon
1919:Main article:
1916:
1913:
1900:
1880:
1875:
1872:
1868:
1864:
1861:
1857:
1853:
1850:
1830:
1810:
1775:
1755:
1729:
1724:
1698:
1695:
1692:
1681:convex polygon
1666:
1661:
1656:
1653:
1638:Main article:
1627:
1624:
1622:
1619:
1610:
1607:
1594:
1591:
1583:
1582:
1570:
1550:
1530:
1513:
1501:
1481:
1461:
1458:
1455:
1435:
1415:
1403:non-decreasing
1398:
1386:
1366:
1342:
1339:
1337:
1334:
1330:Choquet theory
1309:Main article:
1306:
1305:Extreme points
1303:
1276:
1275:
1263:
1254:
1250:
1246:
1243:
1239:
1234:
1231:
1228:
1223:
1218:
1215:
1212:
1209:
1206:
1202:
1166:
1163:
1150:
1147:
1136:cross-polytope
1123:
1103:
1100:
1080:
1060:
1029:
1005:
985:
961:
926:
923:
921:
918:
912:
909:
896:
876:
848:
845:
842:
839:
836:
816:
796:
793:
790:
770:
750:
730:
703:
683:
663:
643:
623:
600:
580:
560:
551:that contains
540:
520:
500:
480:
460:
457:
413:
393:
369:
346:
345:
334:
319:
308:
293:
282:
271:
260:
236:
208:
205:
149:
146:
143:
140:
137:
134:
131:
96:extreme points
65:convex closure
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7480:
7469:
7466:
7464:
7461:
7459:
7456:
7454:
7451:
7449:
7446:
7445:
7443:
7428:
7425:
7424:
7422:
7418:
7412:
7409:
7407:
7404:
7402:
7399:
7397:
7394:
7393:
7391:
7389:
7385:
7378:
7376:
7370:
7368:
7362:
7358:
7354:
7350:
7347:
7346:
7344:
7340:
7334:
7331:
7329:
7325:
7322:
7320:
7317:
7315:
7312:
7310:
7307:
7305:
7302:
7300:
7297:
7295:
7291:
7287:
7283:
7281:
7278:
7277:
7275:
7271:
7265:
7262:
7260:
7257:
7255:
7252:
7250:
7247:
7245:
7244:Mazur's lemma
7242:
7240:
7237:
7235:
7232:
7230:
7227:
7225:
7222:
7220:
7217:
7215:
7212:
7210:
7207:
7206:
7204:
7202:
7197:
7191:
7190:Subderivative
7188:
7186:
7183:
7181:
7178:
7176:
7173:
7171:
7167:
7164:
7162:
7159:
7157:
7154:
7152:
7149:
7147:
7144:
7142:
7138:
7136:
7133:
7131:
7128:
7127:
7125:
7121:
7115:
7112:
7110:
7107:
7105:
7102:
7100:
7097:
7095:
7092:
7090:
7087:
7085:
7082:
7080:
7077:
7075:
7072:
7071:
7069:
7067:
7066:Topics (list)
7063:
7057:
7054:
7052:
7049:
7047:
7044:
7043:
7041:
7037:
7033:
7029:
7022:
7017:
7015:
7010:
7008:
7003:
7002:
6999:
6992:
6988:
6984:
6983:"Convex Hull"
6981:
6977:
6976:
6971:
6970:"Convex Hull"
6967:
6962:
6958:
6954:
6953:
6948:
6947:"Convex hull"
6944:
6943:
6939:
6933:
6929:
6925:
6921:
6917:
6913:
6912:
6906:
6903:
6899:
6895:
6891:
6887:
6883:
6879:
6874:
6871:
6867:
6863:
6859:
6855:
6851:
6847:
6843:
6842:
6836:
6833:
6829:
6825:
6821:
6816:
6813:
6809:
6804:
6799:
6795:
6791:
6790:
6784:
6781:
6777:
6772:
6767:
6763:
6759:
6758:
6753:
6749:
6745:
6741:
6737:
6733:
6728:
6724:
6720:
6716:
6713:
6709:
6705:
6701:
6697:
6692:
6689:
6685:
6681:
6677:
6673:
6669:
6666:(144): 1–40,
6665:
6661:
6660:
6655:
6651:
6648:
6644:
6640:
6636:
6631:
6626:
6622:
6618:
6617:
6612:
6608:
6604:
6602:
6598:
6594:
6589:
6585:
6581:
6576:
6573:
6569:
6565:
6561:
6557:
6553:
6548:
6543:
6539:
6535:
6534:
6528:
6524:
6518:
6511:
6507:
6503:
6501:0-521-35220-7
6497:
6493:
6489:
6485:
6484:
6479:
6475:
6472:
6468:
6464:
6460:
6459:
6453:
6450:
6446:
6442:
6438:
6437:
6432:
6429:; Ruts, Ida;
6428:
6424:
6421:
6417:
6413:
6409:
6405:
6401:
6397:
6393:
6392:
6387:
6383:
6380:
6376:
6372:
6368:
6364:
6361:
6355:
6351:
6350:
6345:
6341:
6338:
6334:
6330:
6326:
6321:
6316:
6312:
6308:
6307:
6301:
6298:
6294:
6290:
6286:
6282:
6278:
6273:
6270:
6264:
6260:
6256:
6252:
6248:
6244:
6241:
6237:
6233:
6231:3-540-40714-6
6227:
6223:
6219:
6215:
6211:
6206:
6203:
6199:
6195:
6191:
6190:
6185:
6180:
6177:
6173:
6168:
6163:
6159:
6155:
6150:
6147:
6143:
6138:
6133:
6129:
6125:
6124:
6118:
6115:
6111:
6107:
6103:
6099:
6095:
6091:
6087:
6082:
6079:
6073:
6069:
6065:
6061:
6056:
6052:
6048:
6044:
6043:Newton, Isaac
6040:
6037:
6033:
6029:
6025:
6021:
6017:
6016:
6011:
6006:
6003:, p. 698
6002:
6001:
5995:
5992:
5988:
5984:
5980:
5976:
5972:
5968:
5964:
5960:
5956:
5953:
5949:
5945:
5943:0-471-09584-2
5939:
5935:
5930:
5927:
5923:
5919:
5915:
5910:
5907:
5903:
5899:
5895:
5890:
5885:
5881:
5877:
5873:
5869:
5868:
5863:
5859:
5855:
5850:
5846:
5842:
5841:
5836:
5832:
5831:Milman, David
5828:
5824:
5822:
5818:
5814:
5809:
5805:
5801:
5797:
5793:
5789:
5788:
5783:
5779:
5770:on 2017-06-20
5769:
5765:
5761:
5757:
5753:
5749:
5747:3-540-55611-7
5743:
5739:
5735:
5731:
5730:
5725:
5721:
5718:
5714:
5709:
5704:
5700:
5696:
5695:
5689:
5686:
5682:
5678:
5674:
5670:
5666:
5661:
5656:
5653:(1): 95–102,
5652:
5648:
5647:
5641:
5638:
5634:
5629:
5624:
5620:
5616:
5612:
5608:
5604:
5600:
5595:
5592:
5590:9780080540221
5586:
5582:
5577:
5573:
5568:
5565:
5561:
5557:
5553:
5548:
5543:
5539:
5535:
5534:
5528:
5525:
5521:
5516:
5511:
5507:
5503:
5502:
5497:
5493:
5490:
5486:
5481:
5476:
5472:
5468:
5467:
5462:
5458:
5456:
5451:
5447:
5443:
5437:
5433:
5429:
5425:
5420:
5411:on 2021-02-28
5407:
5403:
5399:
5395:
5388:
5383:
5380:
5376:
5371:
5366:
5362:
5358:
5357:
5351:
5348:
5346:9780387004242
5342:
5338:
5337:
5332:
5328:
5325:
5321:
5317:
5313:
5309:
5305:
5304:
5299:
5295:
5291:
5289:
5285:
5284:
5277:
5273:
5269:
5265:
5262:
5258:
5254:
5250:
5246:
5242:
5238:
5234:
5233:
5225:
5220:
5217:
5213:
5209:
5207:0-8176-3660-9
5203:
5199:
5195:
5191:
5187:
5183:
5179:
5175:
5172:
5168:
5164:
5160:
5156:
5152:
5148:
5144:
5140:
5136:
5135:
5129:
5126:
5124:9781108641449
5120:
5116:
5112:
5108:
5107:
5101:
5098:
5094:
5090:
5086:
5082:
5078:
5071:
5066:
5063:
5059:
5055:
5051:
5047:
5043:
5039:
5036:
5032:
5028:
5024:
5023:
5018:
5014:
5010:
5006:
5003:
4999:
4995:
4991:
4984:
4980:
4975:
4972:
4968:
4964:
4960:
4956:
4952:
4948:
4944:
4943:
4938:
4934:
4931:
4927:
4923:
4917:
4913:
4909:
4905:
4901:
4897:
4893:
4889:
4886:
4882:
4878:
4874:
4869:
4864:
4860:
4856:
4855:
4849:
4846:
4842:
4838:
4834:
4829:
4826:
4822:
4818:
4814:
4809:
4804:
4800:
4796:
4795:
4787:
4783:
4779:
4776:
4772:
4768:
4764:
4760:
4756:
4755:
4750:
4746:
4743:
4739:
4734:
4729:
4725:
4721:
4720:
4714:
4711:
4707:
4703:
4699:
4695:
4691:
4690:
4685:
4681:
4677:
4673:
4669:
4665:
4661:
4657:
4654:
4650:
4646:
4642:
4638:
4634:
4633:
4627:
4624:
4620:
4616:
4612:
4608:
4604:
4600:
4596:
4595:
4590:
4586:
4583:
4579:
4575:
4571:
4567:
4563:
4558:
4553:
4549:
4545:
4544:
4539:
4535:
4530:
4527:
4523:
4519:
4515:
4510:
4505:
4501:
4497:
4496:
4491:
4487:
4483:
4480:
4476:
4472:
4468:
4464:
4460:
4459:
4454:
4450:
4446:
4443:
4439:
4435:
4431:
4427:
4423:
4419:
4415:
4414:
4406:
4401:
4398:
4394:
4390:
4386:
4382:
4378:
4375:
4371:
4367:
4363:
4362:
4356:
4352:
4351:
4345:
4344:
4339:
4331:
4326:
4323:
4319:
4313:
4310:
4306:
4302:
4298:
4297:Newton (1676)
4293:
4290:
4287:
4283:
4278:
4275:
4271:
4266:
4263:
4259:
4254:
4251:
4247:
4242:
4239:
4235:
4230:
4227:
4223:
4218:
4215:
4211:
4210:Worton (1995)
4206:
4203:
4200:
4196:
4191:
4188:
4184:
4179:
4176:
4172:
4167:
4164:
4160:
4159:Nicola (2000)
4155:
4152:
4148:
4143:
4140:
4136:
4131:
4128:
4124:
4123:Harris (1971)
4119:
4116:
4112:
4107:
4104:
4100:
4095:
4092:
4088:
4083:
4080:
4076:
4071:
4068:
4064:
4059:
4056:
4052:
4047:
4044:
4040:
4035:
4032:
4029:
4025:
4020:
4017:
4013:
4008:
4005:
4001:
3996:
3993:
3989:
3984:
3981:
3977:
3972:
3969:
3965:
3960:
3957:
3953:
3948:
3945:
3941:
3936:
3933:
3929:
3924:
3922:
3918:
3914:
3909:
3906:
3902:
3897:
3895:
3893:
3889:
3885:
3880:
3877:
3873:
3868:
3865:
3861:
3856:
3853:
3849:
3845:
3841:
3836:
3833:
3829:
3825:
3820:
3817:
3814:, p. 13.
3813:
3808:
3805:
3801:
3796:
3793:
3789:
3784:
3781:
3778:, p. 36.
3777:
3772:
3769:
3765:
3764:Seaton (2017)
3760:
3757:
3753:
3748:
3745:
3741:
3740:Sedykh (1981)
3736:
3733:
3729:
3724:
3721:
3717:
3712:
3709:
3705:
3700:
3697:
3693:
3688:
3685:
3681:
3676:
3673:
3670:, p. 57.
3669:
3664:
3661:
3657:
3652:
3649:
3645:
3641:
3636:
3633:
3629:
3624:
3621:
3617:
3612:
3609:
3605:
3600:
3597:
3593:
3589:
3584:
3581:
3577:
3572:
3569:
3566:, Remark 2.6.
3565:
3564:Talman (1977)
3559:
3556:
3552:
3551:Sakuma (1977)
3548:
3544:
3539:
3536:
3533:
3529:
3528:Gustin (1947)
3525:
3520:
3517:
3514:, p. 99.
3513:
3508:
3505:
3501:
3500:Sontag (1982)
3496:
3493:
3489:
3488:Andrew (1979)
3485:
3481:
3476:
3473:
3469:
3465:
3460:
3458:
3454:
3450:
3445:
3442:
3438:
3433:
3430:
3426:
3422:
3418:
3413:
3410:
3406:
3401:
3399:
3397:
3393:
3390:, p. 12.
3389:
3384:
3382:
3378:
3374:
3370:
3366:
3362:
3356:
3353:
3346:
3344:
3342:
3338:
3334:
3331:'s review of
3330:
3326:
3322:
3318:
3314:
3310:
3302:
3300:
3297:
3293:
3289:
3274:compounds. Mg
3273:
3269:
3264:
3257:
3255:
3253:
3249:
3245:
3241:
3233:
3231:
3229:
3228:neighborhoods
3225:
3220:
3216:
3212:
3204:
3202:
3200:
3196:
3192:
3187:
3185:
3181:
3173:
3171:
3169:
3165:
3161:
3157:
3153:
3149:
3145:
3139:
3131:
3129:
3127:
3123:
3119:
3115:
3111:
3107:
3099:
3097:
3095:
3091:
3086:
3084:
3080:
3076:
3068:
3063:
3056:
3054:
3052:
3048:
3044:
3039:
3037:
3033:
3029:
3026:
3022:
3018:
3014:
3010:
3006:
3001:
2999:
2995:
2991:
2987:
2983:
2979:
2975:
2971:
2970:normal matrix
2967:
2963:
2958:
2956:
2952:
2948:
2944:
2940:
2936:
2930:
2925:
2918:
2916:
2914:
2910:
2909:BĂ©zier curves
2906:
2902:
2898:
2894:
2890:
2886:
2882:
2878:
2874:
2870:
2866:
2862:
2854:
2852:
2850:
2846:
2841:
2838:
2837:convex layers
2834:
2829:
2813:
2808:
2805:
2802:
2773:
2758:
2754:
2750:
2742:
2738:
2734:
2730:
2727:
2723:
2719:
2715:
2712:
2708:
2705:
2701:
2697:
2693:
2678:
2674:
2670:
2662:
2659:
2643:
2623:
2615:
2611:
2608:
2604:
2601:
2597:
2594:
2590:
2589:
2588:
2579:
2573:
2565:
2559:
2547:
2545:
2543:
2539:
2535:
2530:
2526:
2522:
2520:
2496:
2492:
2488:
2481:
2474:
2454:
2451:
2448:
2440:
2436:
2417:
2414:
2411:
2408:
2402:
2394:
2375:
2372:
2369:
2366:
2360:
2340:
2332:
2316:
2296:
2276:
2267:
2265:
2261:
2257:
2253:
2249:
2245:
2239:
2231:
2229:
2227:
2223:
2207:
2200:majorized by
2199:
2183:
2175:
2159:
2151:
2145:
2137:
2135:
2133:
2129:
2125:
2121:
2117:
2113:
2105:
2100:
2093:
2091:
2077:
2074:
2070:
2066:
2063:
2060:
2052:
2036:
2016:
2013:
2010:
2007:
2004:
2000:
1996:
1989:in the range
1976:
1968:
1964:
1956:
1954:
1952:
1948:
1944:
1940:
1936:
1927:
1922:
1914:
1912:
1898:
1870:
1866:
1862:
1855:
1848:
1828:
1808:
1800:
1795:
1793:
1789:
1773:
1753:
1745:
1727:
1712:
1696:
1693:
1690:
1682:
1664:
1654:
1651:
1641:
1632:
1625:
1621:Special cases
1620:
1618:
1616:
1608:
1606:
1604:
1600:
1599:Minkowski sum
1593:Minkowski sum
1592:
1590:
1588:
1568:
1548:
1528:
1520:
1519:
1514:
1499:
1479:
1459:
1456:
1453:
1433:
1413:
1405:
1404:
1399:
1384:
1364:
1356:
1352:
1351:
1350:
1348:
1340:
1335:
1333:
1331:
1327:
1323:
1318:
1317:extreme point
1312:
1304:
1302:
1300:
1299:weak topology
1296:
1292:
1287:
1285:
1281:
1261:
1252:
1248:
1244:
1241:
1237:
1232:
1229:
1226:
1213:
1210:
1207:
1200:
1192:
1191:
1190:
1188:
1180:
1176:
1171:
1164:
1162:
1148:
1145:
1137:
1121:
1101:
1098:
1078:
1058:
1049:
1047:
1043:
1027:
1019:
1016:is already a
1003:
983:
975:
959:
950:
948:
944:
940:
936:
932:
924:
919:
917:
910:
908:
894:
874:
866:
862:
843:
840:
837:
814:
794:
791:
788:
768:
748:
728:
720:
715:
701:
681:
661:
641:
621:
612:
598:
578:
558:
538:
518:
498:
478:
465:
458:
456:
454:
450:
449:affine spaces
446:
442:
438:
433:
431:
427:
426:spanning tree
411:
391:
383:
367:
359:
356:with minimum
355:
351:
332:
324:
320:
306:
299:of points in
298:
294:
280:
272:
258:
250:
249:
248:
234:
226:
222:
213:
206:
204:
202:
198:
194:
190:
189:convex layers
186:
182:
178:
174:
170:
165:
163:
144:
141:
138:
135:
129:
121:
117:
113:
109:
105:
101:
97:
93:
89:
84:
82:
78:
74:
70:
66:
62:
58:
54:
46:
41:
37:
33:
19:
7448:Convex hulls
7411:Weak duality
7374:
7366:
7286:Orthogonally
7279:
6973:
6950:
6915:
6909:
6877:
6845:
6839:
6823:
6819:
6793:
6787:
6761:
6755:
6743:
6722:
6703:
6699:
6663:
6657:
6654:Steinitz, E.
6620:
6614:
6592:
6579:
6537:
6531:
6482:
6462:
6456:
6440:
6434:
6395:
6389:
6370:
6348:
6310:
6304:
6280:
6276:
6250:
6213:
6193:
6187:
6184:Wood, Derick
6157:
6153:
6127:
6121:
6089:
6085:
6059:
6050:
6019:
6013:
5999:
5969:(2): 87–98,
5966:
5962:
5933:
5917:
5913:
5871:
5865:
5844:
5838:
5791:
5785:
5782:KĹ‘nig, DĂ©nes
5772:, retrieved
5768:the original
5728:
5698:
5692:
5650:
5644:
5605:(1): 20253,
5602:
5598:
5580:
5571:
5537:
5531:
5508:(1): 89–94,
5505:
5499:
5470:
5464:
5423:
5413:, retrieved
5406:the original
5393:
5360:
5354:
5335:
5307:
5301:
5281:
5275:
5271:
5236:
5230:
5189:
5138:
5132:
5105:
5080:
5076:
5053:
5026:
5020:
4993:
4989:
4946:
4940:
4937:Dines, L. L.
4903:
4858:
4852:
4839:(1): 29–39,
4836:
4832:
4798:
4792:
4758:
4752:
4723:
4717:
4693:
4687:
4675:
4636:
4630:
4598:
4592:
4547:
4541:
4499:
4493:
4486:Bárány, Imre
4462:
4456:
4417:
4411:
4388:
4365:
4359:
4349:
4330:Dines (1938)
4325:
4318:White (1923)
4312:
4292:
4286:Fultz (2020)
4277:
4270:Gibbs (1873)
4265:
4253:
4241:
4229:
4217:
4205:
4190:
4183:Mason (1908)
4178:
4166:
4154:
4147:Katoh (1992)
4142:
4130:
4118:
4106:
4099:Weeks (1993)
4094:
4082:
4070:
4058:
4046:
4034:
4024:Artin (1967)
4019:
4007:
3995:
3988:Brown (1979)
3983:
3976:Rossi (1961)
3971:
3959:
3947:
3935:
3908:
3879:
3867:
3855:
3835:
3819:
3807:
3795:
3783:
3771:
3759:
3747:
3735:
3723:
3711:
3699:
3687:
3675:
3663:
3651:
3635:
3623:
3611:
3599:
3583:
3571:
3558:
3538:
3519:
3507:
3495:
3475:
3449:Knuth (1992)
3444:
3432:
3425:MathOverflow
3412:
3407:, p. 3.
3368:
3360:
3355:
3309:Isaac Newton
3306:
3285:
3237:
3208:
3188:
3184:BĂ©zier curve
3177:
3141:
3103:
3087:
3072:
3047:space curves
3040:
3009:ideal points
3002:
2959:
2933:
2858:
2855:Applications
2845:convex skull
2842:
2833:circumradius
2828:alpha shapes
2746:
2726:metric space
2607:conical hull
2585:
2523:
2268:
2264:face lattice
2241:
2147:
2109:
2094:Space curves
1960:
1946:
1942:
1932:
1796:
1643:
1612:
1596:
1584:
1516:
1401:
1354:
1344:
1314:
1295:Banach space
1288:
1277:
1184:
1050:
951:
938:
930:
928:
914:
716:
613:
491:, for every
470:
434:
350:bounded sets
347:
218:
201:convex skull
177:space curves
166:
102:, and every
92:compact sets
85:
64:
60:
56:
50:
36:
7401:Duality gap
7396:Dual system
7280:Convex hull
6386:Rossi, Hugo
6214:Polynomials
6010:Avis, David
5847:: 133–138,
5827:Krein, Mark
4660:de Berg, M.
4550:(1): 1–28,
4490:Pach, János
4449:Avis, David
4381:Artin, Emil
4320:, page 520.
4316:See, e.g.,
4301:Auel (2019)
4075:Reay (1979)
3860:Chan (2012)
3604:Okon (2000)
3484:Graham scan
3341:Lloyd Dines
3244:state space
3230:of points.
3191:chain girth
3152:budget sets
3083:Tukey depth
2974:eigenvalues
2939:polynomials
2919:Mathematics
2881:Tukey depth
2865:eigenvalues
2861:polynomials
2614:visual hull
2600:linear hull
2593:affine hull
2519:linear time
2331:Graham scan
2232:Computation
2228:operation.
2116:developable
2112:space curve
1587:antimatroid
1046:compact set
976:containing
974:half-spaces
382:rubber band
360:containing
207:Definitions
116:half-spaces
108:algorithmic
104:antimatroid
57:convex hull
18:Convex span
7442:Categories
7324:Radial set
7294:Convex set
7056:Convex set
6911:Biometrics
6547:1603.08409
5959:Lee, D. T.
5821:48.0835.01
5774:2011-09-15
5415:2020-01-01
5141:(1): 171,
5046:Marden, A.
4340:References
3848:Lee (1983)
3592:Lay (1982)
3547:Lay (1982)
3468:Lay (1982)
3373:Fan (1959)
3215:home range
3199:skin girth
3164:non-convex
3057:Statistics
2986:C*-algebra
2947:asymptotic
2913:home range
1821:points in
1518:idempotent
1114:points of
1042:finite set
1018:closed set
807:points in
571:, because
69:convex set
45:convex set
7309:Hypograph
6975:MathWorld
6957:EMS Press
6894:1853/3736
6727:CiteSeerX
6688:122998337
6337:121352925
5862:Krein, M.
5808:128041360
5542:CiteSeerX
5288:pp. 33–54
5278:: 382–404
5232:Ecography
5171:119501393
5097:221659506
4803:CiteSeerX
4552:CiteSeerX
3830:, p. 256.
3549:, p. 21;
3545:, p. 16;
3466:, p. 12;
3268:magnesium
3132:Economics
3025:canonical
2863:, matrix
2679:α
2500:⌋
2486:⌊
2415:
2373:
2138:Functions
2128:sphericon
2078:θ
2067:π
2064:−
2037:θ
2017:π
2011:θ
1997:π
1977:θ
1874:⌋
1860:⌊
1655:⊂
1457:⊆
1355:extensive
1233:≥
1227:
358:perimeter
323:simplices
142:
88:open sets
7333:Zonotope
7304:Epigraph
6902:15514388
6832:43432008
6647:18446330
6609:(1982),
6572:84179479
6517:citation
6480:(1993),
6369:(1970),
6297:20137707
6114:30843551
5991:28600832
5833:(1940),
5815:(1922),
5726:(1992),
5685:15995449
5637:31882982
5450:24287952
5333:(2003),
5261:14592779
4981:(1997),
4825:26605267
4784:(1993),
4674:(2008),
4653:44537056
4442:76650751
4383:(1967),
3594:, p. 43.
3470:, p. 17.
3219:Outliers
3211:ethology
3205:Ethology
2704:geodesic
2542:diameter
2437:and the
2174:epigraph
1679:forms a
1187:open set
943:interior
865:triangle
53:geometry
7388:Duality
7290:Pseudo-
7264:Ursescu
7161:Pseudo-
7135:Concave
7114:Simplex
7094:Duality
6993:, 2007.
6959:, 2001
6932:2533254
6870:0835903
6862:2046536
6812:0404097
6780:1241189
6712:0463985
6680:1580890
6639:0644949
6588:0630708
6564:3765242
6510:1216521
6420:0133479
6412:1970292
6379:0274683
6329:0570883
6240:2082772
6176:1768994
6146:2286077
6094:Bibcode
6036:0552534
5983:0724699
5952:0655598
5906:0002009
5898:1968735
5764:5452191
5756:1226891
5717:1881029
5665:Bibcode
5628:6934831
5607:Bibcode
5564:2107032
5524:0460358
5489:1173256
5402:0356305
5379:0020800
5324:0729228
5241:Bibcode
5216:1264417
5163:0722439
5155:2044692
5062:0903852
5052:(ed.),
5002:1622664
4971:1524247
4963:2302604
4930:2405683
4885:0972777
4877:2244202
4775:0798557
4742:0834056
4710:2994585
4623:1501815
4615:1989687
4582:8013433
4574:1670903
4526:0663877
4518:2044407
4479:1447243
4434:3889348
4397:0237460
3375:, p.48.
3335: (
3319: (
3303:History
3142:In the
3079:bagplot
3067:bagplot
2992:, both
2885:bagplot
1943:pockets
941:is the
935:closure
861:simplex
81:bounded
7371:, and
7342:Series
7259:Simons
7166:Quasi-
7156:Proper
7141:Closed
6930:
6900:
6868:
6860:
6830:
6810:
6778:
6729:
6710:
6686:
6678:
6645:
6637:
6586:
6570:
6562:
6508:
6498:
6418:
6410:
6377:
6356:
6335:
6327:
6295:
6265:
6238:
6228:
6174:
6144:
6112:
6074:
6034:
5989:
5981:
5950:
5940:
5904:
5896:
5819:
5806:
5762:
5754:
5744:
5715:
5683:
5635:
5625:
5587:
5562:
5544:
5522:
5487:
5455:p. 143
5453:; see
5448:
5438:
5400:
5377:
5343:
5322:
5259:
5214:
5204:
5169:
5161:
5153:
5121:
5095:
5060:
5000:
4969:
4961:
4928:
4918:
4883:
4875:
4823:
4805:
4773:
4740:
4708:
4651:
4621:
4613:
4580:
4572:
4554:
4524:
4516:
4477:
4440:
4432:
4395:
4299:; see
3325:German
3272:carbon
3242:, the
2976:. The
2964:, the
2867:, and
2755:, the
2256:facets
2049:. The
1744:vertex
1515:It is
1400:It is
1353:It is
225:convex
199:, and
179:, and
55:, the
7199:Main
6928:JSTOR
6898:S2CID
6858:JSTOR
6828:JSTOR
6684:S2CID
6643:S2CID
6568:S2CID
6542:arXiv
6408:JSTOR
6333:S2CID
6293:S2CID
6110:S2CID
5987:S2CID
5894:JSTOR
5804:S2CID
5760:S2CID
5681:S2CID
5655:arXiv
5409:(PDF)
5390:(PDF)
5257:S2CID
5227:(PDF)
5167:S2CID
5151:JSTOR
5093:S2CID
5073:(PDF)
4986:(PDF)
4959:JSTOR
4873:JSTOR
4821:S2CID
4789:(PDF)
4649:S2CID
4611:JSTOR
4578:S2CID
4514:JSTOR
4438:S2CID
4408:(PDF)
3347:Notes
3333:KĹ‘nig
3036:knots
2988:. In
2984:in a
2968:of a
2955:roots
2538:width
2124:oloid
2104:oloid
1683:when
1446:with
1040:is a
721:, if
7319:Lens
7273:Sets
7123:Maps
7030:and
6664:1914
6523:link
6496:ISBN
6354:ISBN
6263:ISBN
6226:ISBN
6072:ISBN
5938:ISBN
5742:ISBN
5633:PMID
5585:ISBN
5446:PMID
5436:ISBN
5341:ISBN
5202:ISBN
5119:ISBN
4916:ISBN
3337:1922
3321:1935
3195:hull
3154:and
3108:and
2996:and
2941:and
2899:and
2843:The
2826:The
2753:dual
2747:The
2731:The
2709:The
2694:The
2612:The
2605:The
2598:The
2591:The
2540:and
2452:>
2118:and
2014:<
2008:<
1613:The
1426:and
1173:The
929:The
348:For
195:and
112:dual
7373:(Hw
6985:by
6920:doi
6890:hdl
6882:doi
6850:doi
6798:doi
6766:doi
6668:doi
6625:doi
6597:doi
6552:doi
6488:doi
6467:doi
6445:doi
6400:doi
6315:doi
6285:doi
6255:doi
6218:doi
6198:doi
6162:doi
6132:doi
6128:135
6102:doi
6064:doi
6024:doi
5971:doi
5922:doi
5884:hdl
5876:doi
5849:doi
5817:JFM
5796:doi
5734:doi
5703:doi
5699:354
5673:doi
5623:PMC
5615:doi
5552:doi
5510:doi
5475:doi
5428:doi
5365:doi
5312:doi
5249:doi
5194:doi
5143:doi
5111:doi
5085:doi
5031:doi
4951:doi
4908:doi
4863:doi
4841:doi
4813:doi
4763:doi
4728:doi
4698:doi
4641:doi
4603:doi
4562:doi
4504:doi
4467:doi
4422:doi
4370:doi
3486:by
3311:to
3238:In
3178:In
3146:of
3104:In
3073:In
3030:of
3019:in
2960:In
2412:log
2370:log
2242:In
2102:An
1713:in
1315:An
1071:is
447:or
139:log
63:or
51:In
7444::
7365:(H
7363:,
7359:,
7355:,
7292:)
7288:,
7168:)
7146:K-
6989:,
6972:,
6968:,
6955:,
6949:,
6926:,
6916:51
6914:,
6896:,
6888:,
6866:MR
6864:,
6856:,
6846:97
6844:,
6824:17
6822:,
6808:MR
6806:,
6794:38
6792:,
6776:MR
6774:,
6762:52
6760:,
6725:,
6708:MR
6704:29
6702:,
6698:,
6682:,
6676:MR
6674:,
6662:,
6641:,
6635:MR
6633:,
6621:98
6619:,
6613:,
6584:MR
6566:,
6560:MR
6558:,
6550:,
6538:11
6536:,
6519:}}
6515:{{
6506:MR
6504:,
6463:14
6461:,
6441:53
6439:,
6416:MR
6414:,
6406:,
6396:74
6375:MR
6331:,
6325:MR
6323:,
6311:34
6309:,
6291:,
6281:11
6279:,
6261:,
6236:MR
6234:,
6224:,
6212:,
6194:33
6192:,
6172:MR
6170:,
6158:19
6156:,
6142:MR
6140:,
6126:,
6108:,
6100:,
6090:23
6088:,
6070:,
6049:,
6032:MR
6030:,
6018:,
5985:,
5979:MR
5977:,
5967:12
5965:,
5948:MR
5946:,
5918:16
5916:,
5902:MR
5900:,
5892:,
5882:,
5872:41
5843:,
5837:,
5829:;
5802:,
5792:14
5790:,
5758:,
5752:MR
5750:,
5740:,
5713:MR
5711:,
5697:,
5679:,
5671:,
5663:,
5651:19
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5631:,
5621:,
5613:,
5601:,
5560:MR
5558:,
5550:,
5538:30
5536:,
5520:MR
5518:,
5506:15
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5471:44
5469:,
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5434:,
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5392:,
5375:MR
5373:,
5361:53
5359:,
5320:MR
5318:,
5306:,
5296:;
5274:,
5255:,
5247:,
5237:27
5235:,
5229:,
5212:MR
5210:,
5200:,
5184:;
5180:;
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5157:,
5149:,
5139:90
5137:,
5117:,
5091:,
5081:67
5079:,
5075:,
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5044:;
5027:29
5025:,
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4967:MR
4965:,
4957:,
4947:45
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4926:MR
4924:,
4914:,
4898:;
4881:MR
4879:,
4871:,
4859:17
4857:,
4837:20
4835:,
4819:,
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4797:,
4791:,
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4759:31
4757:,
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4736:,
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4704:,
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4692:,
4670:;
4666:;
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4617:,
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4568:,
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4512:,
4500:86
4498:,
4475:MR
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4461:,
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4430:MR
4428:,
4418:66
4416:,
4410:,
4393:MR
4387:,
4364:,
4284:;
4026:;
3920:^
3891:^
3846:;
3842:;
3826:;
3590:;
3530:;
3526:;
3456:^
3423:,
3395:^
3380:^
3065:A
3053:.
3038:.
2915:.
2521:.
2090:.
1911:.
1794:.
1349::
1181:).
455:.
203:.
191:,
187:,
175:,
171:,
59:,
7379:)
7377:)
7375:x
7369:)
7367:x
7351:(
7326:/
7284:(
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7020:e
7013:t
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6922::
6892::
6884::
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6800::
6768::
6670::
6627::
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6490::
6469::
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5973::
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5554::
5512::
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5276:2
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4953::
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4332:.
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4272:.
4260:.
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3766:.
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3606:.
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3502:.
3490:.
3451:.
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3280:3
3278:C
3276:2
3270:–
2814:.
2809:1
2806:+
2803:n
2798:R
2774:n
2769:R
2728:.
2675:/
2671:1
2644:p
2624:p
2505:)
2497:2
2493:/
2489:d
2482:n
2478:(
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2455:3
2449:d
2421:)
2418:h
2409:n
2406:(
2403:O
2379:)
2376:n
2367:n
2364:(
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2277:n
2208:f
2184:f
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2005:2
2001:/
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1879:)
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1867:/
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1856:n
1852:(
1849:O
1829:d
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1723:R
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1691:d
1665:d
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1581:.
1569:X
1549:X
1529:X
1512:.
1500:Y
1480:X
1460:Y
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1397:.
1385:X
1365:X
1262:}
1253:2
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1245:+
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1217:)
1214:y
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682:X
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559:X
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499:X
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412:S
392:S
368:X
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281:X
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148:)
145:n
136:n
133:(
130:O
47:.
34:.
20:)
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