78:
94:
385:. For this bijection, the Hanner polytopes are assumed to be represented geometrically using coordinates in {0,1,−1} rather than as combinatorial equivalence classes; in particular, there are two different geometric forms of a Hanner polytope even in two dimensions, the square with vertex coordinates (±1,±1) and the diamond with vertex coordinates (0,±1) and (±1,0). Given a
211:(including the polytope itself as a face but not including the empty set). For instance, the cube has 8 vertices, 12 edges, 6 squares, and 1 cube (itself) as faces; 8 + 12 + 6 + 1 = 27 = 3. The Hanner polytopes form an important class of examples for
227:
of the two facets is the whole polytope. As a simple consequence of this fact, all facets of a Hanner polytope have the same number of vertices as each other (half the number of vertices of the whole polytope). However, the facets may not all be isomorphic to each other. For instance, in the
393:
vertices correspond to the unit vectors of the space containing the polytope, and for which two vectors are connected by an edge if their sum lies outside the polytope. He observes that the graphs of Hanner polytopes are cographs, which he characterizes in two ways: the graphs with no
398:
of length three, and the graphs whose induced subgraphs are all either disconnected or the complements of disconnected graphs. Conversely, every cograph can be represented in this way by a Hanner polytope.
115:, is also a Hanner polytope, the direct sum of three line segments. In three dimensions all Hanner polytopes are combinatorially equivalent to one of these two types of polytopes. In higher dimensions the
58:
They are exactly the polytopes that can be constructed using only these rules: that is, every Hanner polytope can be formed from line segments by a sequence of product and dual operations.
65:, the dual of the Cartesian products. This direct sum operation combines two polytopes by placing them in two linearly independent subspaces of a larger space and then constructing the
191:
Because the polar dual of a Hanner polytope is another Hanner polytope, the Hanner polytopes have the property that both they and their duals have coordinates in {0,1,−1}.
471:
444:
799:
670:
312:
later showed that this property can be used to characterize the Hanner polytopes: they are (up to affine transformation) exactly the polytopes for which
51:
The
Cartesian product of every two Hanner polytopes is another Hanner polytope, whose dimension is the sum of the dimensions of the two given polytopes.
358:
236:. Dually, in every Hanner polytope, every two opposite vertices touch disjoint sets of facets, and together touch all of the facets of the polytope.
596:
248:
of a Hanner polytope (the product of its volume and the volume of its polar dual) is the same as for a cube or cross polytope. If the
223:
In a Hanner polytope, every two opposite facets are disjoint, and together include all of the vertices of the polytope, so that the
644:
Yaroslavl
International Conference "Discrete Geometry" dedicated to the centenary of A.D.Alexandrov (Yaroslavl, August 13-18, 2012)
641:
Kozachok, Marina (2012), "Perfect prismatoids and the conjecture concerning with face numbers of centrally symmetric polytopes",
840:
Hansen, Allan B.; Lima, Ȧsvald (1981), "The structure of finite-dimensional Banach spaces with the 3.2. intersection property",
642:
276:: every set of translates that have nonempty pairwise intersections has a nonempty intersection. Moreover, these are the only
61:
Alternatively and equivalently to the polar dual operation, the Hanner polytopes may be constructed by
Cartesian products and
123:, analogues of the cube and octahedron, are again Hanner polytopes. However, more examples are possible. For instance, the
819:
708:
Martini, H.; Swanepoel, K. J.; de Wet, P. Oloff (2009), "Absorbing angles, Steiner minimal trees, and antipodality",
156:
are Hanner polytopes with coordinates in this form, then the coordinates of the vertices of the
Cartesian product of
212:
336:
552:
668:
Reisner, S. (1991), "Certain Banach spaces associated with graphs and CL-spaces with 1-unconditional bases",
389:-dimensional polytope with vertex coordinates in {0,1,−1}, Reisner defines an associated graph whose
111:
is a Hanner polytope, and can be constructed as a
Cartesian product of three line segments. Its dual, the
300:
that do not form a Helly family (they intersect pairwise but have an empty intersection). He showed that
308:) is either three or four, and gave the Hanner polytopes as examples of polytopes for which it is four.
148:
Every Hanner polytope can be given vertex coordinates that are 0, 1, or −1. More explicitly, if
591:
449:
524:
184:
with a vector of zeros, or by concatenating a vector of zeros with the coordinates of a vertex in
885:
762:
717:
650:, P.G. Demidov Yaroslavl State University, International B.N. Delaunay Laboratory, pp. 46–49
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422:
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is true, these polytopes are the minimizers of Mahler volume among all the centrally symmetric
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132:
25:
851:
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29:
21:
419:. The Hanner spaces are the spaces that can be built up from one-dimensional spaces by
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204:
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532:, Ph.D. thesis, Department of Mathematical Sciences, Chalmers Institute of Technology
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77:
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66:
33:
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with an octahedron as its base, is also a Hanner polytope, as is its dual, the
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The dual of a Hanner polytope is another Hanner polytope of the same dimension.
776:
731:
619:
112:
86:
62:
547:
408:
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594:(2009), "On Kalai's conjectures concerning centrally symmetric polytopes",
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The Hanner polytopes are constructed recursively by the following rules:
856:
565:
374:
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that all centrally symmetric polytopes have at least 3 nonempty faces.
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with this property. For any other centrally symmetric convex polytope
493:
Hanner, Olof (1956), "Intersections of translates of convex bodies",
753:
Kim, Jaegil (2014), "Minimal volume product near Hanner polytopes",
328:
The number of combinatorial types of Hanner polytopes of dimension
767:
722:
610:
232:, two of the facets are octahedra, and the other eight facets are
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are formed either by concatenating the coordinates of a vertex in
92:
76:
550:(1989), "The number of faces of centrally-symmetric polytopes",
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82:
526:
Topics in algorithmic, enumerative and geometric combinatorics
353:
164:
are formed by concatenating the coordinates of a vertex in
172:. The coordinates of the vertices of the direct sum of
800:"Ein Satz über Parallelverschiebungen konvexer Körper"
452:
425:
48:
A line segment is a one-dimensional Hanner polytope.
351:1, 1, 2, 4, 8, 18, 40, 94, 224, 548, ... (sequence
465:
438:
710:Journal of Optimization Theory and Applications
296:) to be the smallest number of translates of
32:operations. Hanner polytopes are named after
8:
89:, the two three-dimensional Hanner polytopes
671:Journal of the London Mathematical Society
369:between the Hanner polytopes of dimension
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855:
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268:(or of an affine transformation of it, a
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219:Pairs of opposite facets and vertices
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168:with the coordinates of a vertex in
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411:of a family of finite-dimensional
14:
804:Acta Universitatis Szegediensis
466:{\displaystyle \ell _{\infty }}
36:, who introduced them in 1956.
755:Journal of Functional Analysis
1:
590:Sanyal, Raman; Werner, Axel;
407:The Hanner polytopes are the
332:is the same as the number of
24:constructed recursively by
902:
777:10.1016/j.jfa.2013.08.008
732:10.1007/s10957-009-9552-1
620:10.1007/s00454-008-9104-8
439:{\displaystyle \ell _{1}}
324:Combinatorial enumeration
199:Every Hanner polytope is
144:Coordinate representation
684:10.1112/jlms/s2-43.1.137
553:Graphs and Combinatorics
495:Mathematica Scandinavica
310:Hansen & Lima (1981)
798:Sz.-Nagy, Béla (1954),
523:Freij, Ragnar (2012),
467:
440:
347:= 1, 2, 3, ... it is:
337:series–parallel graphs
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81:The three-dimensional
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381:vertices is given by
343:unlabeled edges. For
127:, a four-dimensional
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450:
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320:) > 3.
264:The translates of a
213:Kalai's 3 conjecture
201:centrally symmetric
857:10.1007/BF02392457
592:Ziegler, Günter M.
566:10.1007/BF01788696
463:
436:
203:, and has exactly
105:
91:
85:and its dual, the
674:, Second Series,
250:Mahler conjecture
234:triangular prisms
133:cubical bipyramid
26:Cartesian product
893:
870:
868:
859:
843:Acta Mathematica
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365:A more explicit
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230:octahedral prism
125:octahedral prism
102:octahedral prism
98:Schlegel diagram
69:of their union.
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195:Number of faces
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22:convex polytope
18:Hanner polytope
16:In geometry, a
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604:(2): 183–198,
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473:combinations.
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383:Reisner (1991)
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850:(1–2): 1–23,
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822:on 2016-03-04
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417:Hanner spaces
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413:Banach spaces
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246:Mahler volume
240:Mahler volume
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396:induced path
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274:Helly family
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40:Construction
17:
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810:: 169–177,
225:convex hull
67:convex hull
63:direct sums
34:Olof Hanner
826:2013-05-19
548:Kalai, Gil
477:References
409:unit balls
139:Properties
117:hypercubes
113:octahedron
87:octahedron
30:polar dual
886:Polytopes
768:1212.2544
723:1108.5046
611:0708.3661
501:: 65–87,
459:∞
455:ℓ
428:ℓ
367:bijection
272:) form a
266:hypercube
207:nonempty
880:Category
375:cographs
373:and the
288:defined
73:Examples
866:0594626
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334:simple
763:arXiv
718:arXiv
648:(PDF)
606:arXiv
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377:with
339:with
209:faces
129:prism
20:is a
446:and
359:OEIS
244:The
176:and
160:and
152:and
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109:cube
83:cube
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