62:
says that of all bodies with the same volume, the ball has the smallest surface area. In other manifolds it is usually very difficult to find the precise body minimizing the surface area, and this is not what the isoperimetric dimension is about. The question we will ask is, what is
185:
361:
276:) has topological dimension 2 but isoperimetric dimension 1. Indeed, multiplying any manifold with a compact manifold does not change the isoperimetric dimension (it only changes the value of the constant
448:
481:. In general, the opposite is not true, i.e. even uniformly exponential volume growth does not imply any kind of isoperimetric inequality. A simple example can be had by taking the graph
614:
661:
104:
307:
750:
This paper contains the result that on groups of polynomial growth, volume growth and isoperimetric inequalities are equivalent. In French.
190:
The notations vol and area refer to the regular notions of volume and surface area on the manifold, or more precisely, if the manifold has
283:
It is also possible for the isoperimetric dimension to be larger than the topological dimension. The simplest example is the infinite
714:
785:
772:
This paper contains a precise definition of the isoperimetric dimension of a graph, and establishes many of its properties.
396:
288:
387:
525:
478:
550:
297:
has topological dimension 2 and isoperimetric dimension infinity. In fact the hyperbolic plane has positive
76:
40:
532:
505:
474:
294:
44:
790:
710:
265:
543:
Varopoulos' theorem: If G is a graph satisfying a d-dimensional isoperimetric inequality then
624:
298:
501:|. Both properties (exponential growth and 0 isoperimetric dimension) are easy to verify.
55:
36:
180:{\displaystyle \operatorname {area} (\partial D)\geq C\operatorname {vol} (D)^{(d-1)/d}.}
779:
521:
254:
59:
536:
269:
20:
356:{\displaystyle \operatorname {area} (\partial D)\geq C\operatorname {vol} (D),}
284:
707:
Isoperimetric
Inequalities: Differential geometric and analytic perspectives
517:
273:
535:
connects the isoperimetric dimension of a graph to the rate of escape of
28:
764:
287:, which has topological dimension 2 and isoperimetric dimension 3. See
721:
Discusses the topic in the context of manifolds, no mention of graphs.
67:
the minimal surface area, whatever the body realizing it might be.
257:— as discussed above, for the Euclidean space the constant
261:
is known precisely since the minimum is achieved for the ball.
508:. It turns out that a group with polynomial growth of order
280:). Any compact manifold has isoperimetric dimension 0.
366:
which obviously implies infinite isoperimetric dimension.
249:-dimensional Euclidean space has isoperimetric dimension
206:
here refers to some constant, which does not depend on
31:
is a notion of dimension that tries to capture how the
627:
553:
443:{\displaystyle \operatorname {vol} B(x,r)\geq Cr^{d}}
399:
310:
107:
709:, Cambridge university press, Cambridge, UK (2001),
655:
608:
442:
355:
179:
382:-dimensional isoperimetric inequality implies a
739:Isopérimétrie pour les groupes et les variétés
301:. This means that it satisfies the inequality
493: + 1) and connecting to the vertex
8:
728:Isoperimetric inequalities and Markov chains
378:(or sum in the case of graphs) shows that a
737:Thierry Coulhon and Laurent Saloff-Coste,
485:(i.e. all the integers with edges between
202: − 1)-dimensional volume.
194:topological dimensions then vol refers to
632:
626:
596:
589:
558:
552:
434:
398:
309:
164:
148:
106:
664:is the probability that a random walk on
609:{\displaystyle p_{n}(x,y)\leq Cn^{-d/2}}
504:An interesting exception is the case of
198:-dimensional volume and area refers to (
237:-dimensional isoperimetric inequality.
51:against those of the Euclidean space).
763:, International Press, (2004), 53–82.
210:(it may depend on the manifold and on
7:
765:http://math.ucsd.edu/~fan/wp/iso.pdf
35:of the manifold resembles that of a
761:Surveys in Differential Geometry IX
757:Discrete Isoperimetric Inequalities
290:for pictures and Mathematica code.
516:. This holds both for the case of
497:a complete binary tree of height |
320:
117:
14:
225:is the supremum of all values of
539:on the graph. The result states
98:with a smooth boundary one has
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638:
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564:
421:
409:
347:
341:
326:
317:
161:
149:
145:
138:
123:
114:
1:
465:) denotes the ball of radius
264:An infinite cylinder (i.e. a
512:has isoperimetric dimension
370:Consequences of isoperimetry
741:, Rev. Mat. Iberoamericana
807:
374:A simple integration over
253:. This is the well known
526:finitely generated group
88:isoperimetric inequality
60:isoperimetric inequality
47:which compare different
656:{\textstyle p_{n}(x,y)}
219:isoperimetric dimension
77:differentiable manifold
25:isoperimetric dimension
657:
610:
444:
357:
181:
786:Mathematical analysis
658:
611:
445:
358:
255:isoperimetric problem
182:
41:topological dimension
625:
551:
397:
308:
105:
90:if for any open set
82:that it satisfies a
33:large-scale behavior
726:N. Th. Varopoulos,
475:Riemannian distance
45:Hausdorff dimension
16:Concept in topology
730:, J. Funct. Anal.
653:
606:
440:
353:
177:
694:is some constant.
469:around the point
71:Formal definition
798:
745:(1993), 293–314.
734:(1985), 215–239.
662:
660:
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612:
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600:
563:
562:
449:
447:
446:
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438:
362:
360:
359:
354:
299:Cheeger constant
295:hyperbolic plane
186:
184:
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178:
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172:
168:
806:
805:
801:
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799:
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623:
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430:
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103:
102:
75:We say about a
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56:Euclidean space
49:local behaviors
37:Euclidean space
17:
12:
11:
5:
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802:
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788:
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769:
768:
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747:
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723:
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705:Isaac Chavel,
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635:
631:
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479:graph distance
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72:
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15:
13:
10:
9:
6:
4:
3:
2:
803:
792:
789:
787:
784:
783:
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771:
770:
766:
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754:
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749:
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736:
733:
729:
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724:
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715:0-521-80267-9
712:
708:
704:
703:
698:
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677:
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670:starting from
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633:
629:
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582:
579:
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540:
538:
534:
531:A theorem of
529:
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507:
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476:
472:
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456:
435:
431:
427:
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418:
415:
412:
406:
403:
400:
393:
392:
391:
389:
388:volume growth
386:-dimensional
385:
381:
377:
369:
367:
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344:
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329:
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314:
311:
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135:
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111:
108:
101:
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86:-dimensional
85:
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70:
68:
66:
65:approximately
61:
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52:
50:
46:
42:
38:
34:
30:
26:
22:
760:
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742:
738:
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706:
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675:
672:
669:
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663:
619:
618:
542:
541:
530:
522:Cayley graph
520:and for the
513:
509:
503:
498:
494:
490:
486:
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470:
466:
462:
458:
454:
452:
383:
379:
375:
373:
365:
292:
282:
277:
263:
258:
250:
246:
244:
234:
233:satisfies a
230:
226:
222:
218:
216:
211:
207:
203:
199:
195:
191:
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95:
91:
87:
83:
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64:
53:
48:
39:(unlike the
32:
24:
18:
755:Fan Chung,
537:random walk
21:mathematics
780:Categories
699:References
688:steps, and
676:will be in
533:Varopoulos
518:Lie groups
477:or in the
285:jungle gym
229:such that
791:Dimension
591:−
580:≤
425:≥
404:
390:, namely
339:
330:≥
321:∂
315:
156:−
136:
127:≥
118:∂
112:
272:and the
241:Examples
29:manifold
473:in the
268:of the
266:product
54:In the
43:or the
713:
506:groups
453:where
270:circle
58:, the
23:, the
682:after
620:where
524:of a
27:of a
732:63:2
711:ISBN
489:and
312:area
293:The
274:line
217:The
109:area
743:9:2
401:vol
336:vol
221:of
214:).
133:vol
94:in
19:In
782::
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510:d
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495:n
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471:x
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459:x
457:(
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259:C
251:d
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231:M
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175:.
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