292:
hyperplanes. In particular, in five dimensions, sets of five hyperplanes can partition segments of the moment curve into at most 26 pieces. It is not known whether four-dimensional partitions into 16 equal subsets by four hyperplanes are always possible, but it is possible to partition 16 points on
288:. Similarly but more complicatedly, any volume or measure in three dimensions may be partitioned into eight equal subsets by three planes. However, this result does not generalize to five or more dimensions, as the moment curve provides examples of sets that cannot be partitioned into 2 subsets by
387:
Then a plane can only cross the curve at three positions. Since two crossing edges must have four vertices in the same plane, this can never happen. A similar construction using the moment curve modulo a prime number, but in two dimensions rather than three, provides a linear bound for the
119:
263:
203:. Cyclic polytopes have the largest possible number of faces for a given number of vertices, and in dimensions four or more have the property that their edges form a
743:
714:
688:
215:/2 vertices of the polytope forms one of its faces. Sets of points on the moment curve also realize the maximum possible number of simplices,
728:
704:
718:
646:
544:; Attali, Dominique; Devillers, Olivier (2007), "Complexity of Delaunay triangulation for points on lower-dimensional polyhedra",
183:
points, then the curve crosses the hyperplane at each intersection point. Thus, every finite point set on the moment curve is in
566:
46:
296:
A construction based on the moment curve can be used to prove a lemma of Gale, according to which, for any positive integers
494:, Section 3.5, Gale's Lemma and Schrijver's Theorem, pp. 64–67. The use of Gale's lemma for the coloring problem is due to
696:
218:
774:
675:, Symposia in Pure Mathematics, vol. 7, Providence, R.I.: American Mathematical Society, pp. 225–232,
389:
152:
266:
329:
137:
634:
343:-vertex graphs may be drawn with their vertices in a three-dimensional integer grid of side length O(
285:
37:
208:
724:
700:
642:
561:
144:
752:
720:
Using the Borsuk-Ulam
Theorem: Lectures on Topological Methods in Combinatorics and Geometry
614:
575:
321:
184:
156:
680:
656:
641:, EATCS Monographs on Theoretical Computer Science, vol. 10, Berlin: Springer-Verlag,
626:
587:
553:
676:
652:
622:
583:
549:
281:
200:
148:
125:
33:
25:
204:
768:
579:
524:
336:
293:
the four-dimensional moment curve into the 16 orthants of a set of four hyperplanes.
133:
605:
600:
325:
160:
284:, it is possible to divide any area or measure into four equal subsets, using the
668:
596:
541:
196:
756:
316:-dimensional sphere in such a way that every open hemisphere contains at least
738:
664:
546:
Proceedings of the
Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms
172:
347:) and with no two edges crossing. The main idea is to choose a prime number
129:
17:
618:
320:
points. This lemma, in turn, can be used to calculate the
143:
Moment curves have been used for several applications in
179:
points. If a hyperplane intersects the curve in exactly
114:{\displaystyle \left(x,x^{2},x^{3},\dots ,x^{d}\right).}
175:
intersects the moment curve in a finite set of at most
463:
221:
199:
of any finite set of points on the moment curve is a
49:
507:
257:
113:
564:(1978), "A short proof of Kneser's conjecture",
328:, a problem first solved in a different way by
667:(1963), "Neighborly and cyclic polytopes", in
258:{\displaystyle \Omega (n^{\lceil d/2\rceil })}
8:
475:
447:
423:
247:
233:
603:(1997), "Three-dimensional graph drawing",
744:Journal of the London Mathematical Society
239:
232:
220:
136:. Its closure in projective space is the
132:, and in three-dimensional space it is a
97:
78:
65:
48:
491:
479:
451:
431:
427:
411:
407:
400:
335:The moment curve has also been used in
495:
741:(1951), "On a problem of Heilbronn",
548:, New York: ACM, pp. 1106–1113,
464:Amenta, Attali & Devillers (2007)
7:
639:Algorithms in Combinatorial Geometry
520:
443:
211:, meaning that each set of at most
699:, vol. 212, Springer-Verlag,
222:
14:
36:given by the set of points with
567:Journal of Combinatorial Theory
155:, and a geometric proof of the
252:
225:
1:
697:Graduate Texts in Mathematics
693:Lectures on Discrete Geometry
580:10.1016/0097-3165(78)90023-7
359:of the graph at coordinates
410:, Definition 5.4.1, p. 97;
304:, it is possible to place 2
791:
723:, Universitext, Springer,
414:, Definition 1.6.3, p. 17.
207:. More strongly, they are
757:10.1112/jlms/s1-26.3.198
673:Convexity, Seattle, 1961
390:no-three-in-line problem
153:no-three-in-line problem
128:, the moment curve is a
267:Delaunay triangulations
185:affine general position
430:, Lemma 5.4.2, p. 97;
259:
115:
635:Edelsbrunner, Herbert
454:, Lemma 5.4.2, p. 97.
434:, Lemma 1.6.4, p. 17.
265:, among all possible
260:
138:rational normal curve
116:
38:Cartesian coordinates
355:and to place vertex
286:ham sandwich theorem
219:
209:neighborly polytopes
47:
508:Cohen et al. (1997)
476:Edelsbrunner (1987)
448:Edelsbrunner (1987)
424:Edelsbrunner (1987)
339:, to show that all
619:10.1007/BF02522826
255:
111:
730:978-3-540-00362-5
706:978-0-387-95373-1
145:discrete geometry
782:
775:Algebraic curves
759:
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322:chromatic number
264:
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157:chromatic number
149:cyclic polytopes
120:
118:
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107:
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83:
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70:
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790:
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713:
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492:Matoušek (2003)
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480:Matoušek (2003)
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452:Matoušek (2002)
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432:Matoušek (2003)
428:Matoušek (2002)
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412:Matoušek (2003)
408:Matoušek (2002)
406:
402:
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379: mod
371: mod
282:Euclidean plane
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217:
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201:cyclic polytope
193:
169:
126:Euclidean plane
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61:
54:
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34:Euclidean space
26:algebraic curve
12:
11:
5:
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762:
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751:(3): 198–204,
735:
729:
715:Matoušek, JiĹ™Ă
711:
705:
689:Matoušek, JiĹ™Ă
685:
661:
647:
631:
613:(2): 199–208,
595:Cohen, R. F.;
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574:(3): 325–326,
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205:complete graph
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648:3-540-13722-X
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496:Bárány (1978)
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488:
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481:
478:, pp. 70–79;
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337:graph drawing
333:
331:
330:László Lovász
327:
326:Kneser graphs
323:
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308: +
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161:Kneser graphs
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134:twisted cubic
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35:
32:-dimensional
31:
27:
23:
19:
748:
742:
719:
692:
672:
669:Klee, Victor
638:
610:
606:Algorithmica
604:
599:; Lin, Tao;
571:
570:, Series A,
565:
545:
542:Amenta, Nina
519:Credited by
515:
503:
487:
482:, pp. 50–51.
471:
459:
439:
419:
403:
386:
380:
376:
372:
368:
364:
356:
352:
351:larger than
348:
344:
340:
334:
317:
313:
312:points on a
309:
305:
301:
297:
295:
289:
279:
277:dimensions.
274:
270:
212:
194:
191:Applications
180:
176:
170:
142:
123:
40:of the form
29:
22:moment curve
21:
15:
739:Roth, K. F.
665:Gale, David
521:Roth (1951)
444:Gale (1963)
269:of sets of
197:convex hull
601:Ruskey, F.
562:Bárány, I.
534:References
525:Paul Erdős
450:, p. 101;
426:, p. 100;
273:points in
173:hyperplane
167:Properties
147:including
597:Eades, P.
248:⌉
234:⌈
223:Ω
88:…
769:Category
717:(2003),
691:(2002),
637:(1987),
130:parabola
18:geometry
681:0152944
671:(ed.),
657:0904271
627:1425733
588:0514626
554:2485262
324:of the
280:In the
124:In the
727:
703:
679:
655:
645:
625:
586:
552:
171:Every
151:, the
24:is an
20:, the
396:Notes
725:ISBN
701:ISBN
643:ISBN
300:and
195:The
753:doi
615:doi
576:doi
523:to
159:of
28:in
16:In
771::
749:26
747:,
695:,
677:MR
653:MR
651:,
623:MR
621:,
611:17
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584:MR
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572:25
550:MR
446:;
392:.
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375:,
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755::
734:.
710:.
684:.
660:.
630:.
617::
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381:p
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341:n
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314:d
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