527:
356:
557:= { (0,0), (1,0), (2,0), (1,1), (2,1) } is integrally convex, and indeed admits an integral triangulation, e.g. with the three simplices {(0,0),(1,0),(1,1)} and {(1,0),(2,0),(2,1)} and {(1,0),(1,1),(2,1)}. See image at the right.
468:
520:
325:
263:
226:
186:
138:
73:
637:
620:
Chen, Xi; Deng, Xiaotie (2006). "A Simplicial
Approach for Discrete Fixed Point Theorems". In Chen, Danny Z.; Lee, D. T. (eds.).
493:
The vertices of every simplex of the triangulation lie in the same "cell" (hypercube of side-length 1) of the integer grid
701:
471:
438:
526:
496:
301:
239:
202:
162:
114:
49:
624:. Lecture Notes in Computer Science. Vol. 4112. Berlin, Heidelberg: Springer. pp. 3–12.
602:
88:
677:
633:
594:
188:, since it contains all the real points that are convex combinations of the integer points in
32:
654:
669:
625:
586:
432:
Iimura, Murota and Tamura have shown the following property of integrally convex set.
355:
695:
606:
17:
99:, where "near" means that the distance between each two coordinates is less than 1.
673:
287:} }. These are the integer points that are considered "nearby" to the real point
577:
Yang, Zaifu (2009-12-01). "Discrete fixed point analysis and its applications".
148:
80:
590:
36:
681:
598:
629:
550:, or has to include simplices that are not contained in a single cell.
542:) does not admit an integral triangulation: every triangulation of ch(
525:
354:
424:= { (0,0), (1,0), (2,0), (1,1), (2,1) } is integrally convex.
653:
Iimura, Takuya; Murota, Kazuo; Tamura, Akihisa (2005-12-01).
486:
The vertices of the triangulation are the vertices of
499:
441:
371:= { (0,0), (1,0), (2,0), (2,1) }. Its convex hull ch(
304:
242:
205:
165:
117:
52:
514:
470:be a finite integrally convex set. There exists a
462:
319:
257:
220:
180:
132:
67:
579:Journal of Fixed Point Theory and Applications
8:
655:"Discrete fixed point theorem reconsidered"
506:
502:
501:
498:
463:{\displaystyle X\subset \mathbb {Z} ^{n}}
454:
450:
449:
440:
311:
307:
306:
303:
249:
245:
244:
241:
212:
208:
207:
204:
172:
168:
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164:
124:
120:
119:
116:
59:
55:
54:
51:
538:is not integrally convex, and indeed ch(
566:
390:) = {(1,0), (2,0), (1,1), (2,1) }. So
546:), either has to add vertices not in
7:
572:
570:
375:) contains, for example, the point
75:is integrally convex if any point
25:
662:Journal of Mathematical Economics
515:{\displaystyle \mathbb {Z} ^{n}}
320:{\displaystyle \mathbb {Z} ^{n}}
258:{\displaystyle \mathbb {Z} ^{n}}
221:{\displaystyle \mathbb {R} ^{n}}
181:{\displaystyle \mathbb {R} ^{n}}
133:{\displaystyle \mathbb {Z} ^{n}}
68:{\displaystyle \mathbb {Z} ^{n}}
398:) = {(1,0), (2,0), (2,1)}. But
1:
674:10.1016/j.jmateco.2005.03.001
410:)). See image at the right.
417:is not integrally convex.
622:Computing and Combinatorics
35:analogue of the concept of
718:
382:The integer points nearby
591:10.1007/s11784-009-0130-9
359:Non-integrally convex set
531:
516:
464:
360:
321:
259:
222:
182:
134:
87:can be expressed as a
69:
553:In contrast, the set
530:Integrally convex set
529:
517:
465:
420:In contrast, the set
358:
322:
260:
223:
183:
135:
70:
29:integrally convex set
18:Integrally-convex set
497:
439:
302:
240:
203:
163:
115:
50:
46:of the integer grid
630:10.1007/11809678_3
532:
512:
460:
361:
317:
255:
218:
178:
130:
89:convex combination
65:
702:Discrete geometry
639:978-3-540-36926-4
329:integrally convex
279:| < 1 for all
159:) is a subset of
91:of the points of
33:discrete geometry
16:(Redirected from
709:
686:
685:
668:(8): 1030–1036.
659:
650:
644:
643:
617:
611:
610:
574:
534:The example set
521:
519:
518:
513:
511:
510:
505:
469:
467:
466:
461:
459:
458:
453:
339:) is also in ch(
326:
324:
323:
318:
316:
315:
310:
264:
262:
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256:
254:
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248:
227:
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187:
185:
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179:
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139:
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131:
129:
128:
123:
95:that are "near"
74:
72:
71:
66:
64:
63:
58:
21:
717:
716:
712:
711:
710:
708:
707:
706:
692:
691:
690:
689:
657:
652:
651:
647:
640:
619:
618:
614:
576:
575:
568:
563:
500:
495:
494:
448:
437:
436:
430:
353:
331:if every point
305:
300:
299:
277:
270:
243:
238:
237:
228:, denote near(
206:
201:
200:
166:
161:
160:
155:. Note that ch(
118:
113:
112:
111:be a subset of
105:
53:
48:
47:
23:
22:
15:
12:
11:
5:
715:
713:
705:
704:
694:
693:
688:
687:
645:
638:
612:
585:(2): 351–371.
565:
564:
562:
559:
524:
523:
509:
504:
491:
457:
452:
447:
444:
429:
426:
379:= (1.2, 0.5).
352:
349:
314:
309:
275:
268:
252:
247:
215:
210:
195:For any point
175:
170:
127:
122:
104:
101:
62:
57:
24:
14:
13:
10:
9:
6:
4:
3:
2:
714:
703:
700:
699:
697:
683:
679:
675:
671:
667:
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656:
649:
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641:
635:
631:
627:
623:
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613:
608:
604:
600:
596:
592:
588:
584:
580:
573:
571:
567:
560:
558:
556:
551:
549:
545:
541:
537:
528:
507:
492:
489:
485:
484:
483:
481:
477:
473:
472:triangulation
455:
445:
442:
433:
427:
425:
423:
418:
416:
411:
409:
405:
402:is not in ch(
401:
397:
393:
389:
385:
380:
378:
374:
370:
366:
357:
350:
348:
346:
342:
338:
334:
330:
312:
297:
292:
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286:
282:
278:
271:
250:
235:
232:) := {
231:
213:
198:
193:
191:
173:
158:
154:
150:
146:
143:Denote by ch(
141:
125:
110:
102:
100:
98:
94:
90:
86:
82:
78:
60:
45:
40:
39:in geometry.
38:
34:
30:
19:
665:
661:
648:
621:
615:
582:
578:
554:
552:
547:
543:
539:
535:
533:
487:
479:
475:
434:
431:
421:
419:
414:
412:
407:
403:
399:
395:
391:
387:
383:
381:
376:
372:
368:
367:= 2 and let
364:
362:
344:
340:
336:
332:
328:
295:
293:
288:
284:
280:
273:
266:
233:
229:
196:
194:
189:
156:
152:
144:
142:
108:
106:
96:
92:
84:
76:
43:
41:
28:
26:
149:convex hull
103:Definitions
81:convex hull
561:References
478:) that is
428:Properties
413:Therefore
327:is called
283:in {1,...,
37:convex set
682:0304-4068
607:122640338
599:1661-7746
446:⊂
386:are near(
294:A subset
42:A subset
696:Category
482:, i.e.:
480:integral
406:∩ near(
394:∩ near(
351:Example
343:∩ near(
79:in the
31:is the
680:
636:
605:
597:
474:of ch(
335:in ch(
147:) the
658:(PDF)
603:S2CID
678:ISSN
634:ISBN
595:ISSN
435:Let
363:Let
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265:| |
107:Let
670:doi
626:doi
587:doi
298:of
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83:of
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581:.
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140:.
684:.
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628::
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536:X
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490:;
488:X
476:X
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415:X
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392:X
388:y
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333:y
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285:n
281:i
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126:n
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93:X
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61:n
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