Editing Isoperimetric dimension

{{Short description|Concept in topology}}
In [[mathematics]], the '''isoperimetric dimension''' of a [[manifold]] is a notion of dimension that tries to capture how the ''large-scale behavior'' of the manifold resembles that of a [[Euclidean space]] (unlike the [[topological dimension]] or the [[Hausdorff dimension]] which compare different ''local behaviors'' against those of the Euclidean space).

In the [[Euclidean space]], the [[isoperimetry|isoperimetric inequality]] 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 ''approximately'' the minimal surface area, whatever the body realizing it might be.

==Formal definition==

We say about a [[differentiable manifold]] ''M'' that it satisfies a ''d''-dimensional '''isoperimetric inequality''' if for any open set ''D'' in ''M'' with a smooth boundary one has

:<math>\operatorname{area}(\partial D)\geq C\operatorname{vol}(D)^{(d-1)/d}.</math>

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 ''n'' topological dimensions then vol refers to ''n''-dimensional volume and area refers to (''n''&nbsp;&minus;&nbsp;1)-dimensional volume. ''C'' here refers to some constant, which does not depend on ''D'' (it may depend on the manifold and on ''d'').

The '''isoperimetric dimension''' of ''M'' is the [[Infimum and supremum|supremum]] of all values of ''d'' such that ''M'' satisfies a ''d''-dimensional isoperimetric inequality.

==Examples==

A ''d''-dimensional Euclidean space has isoperimetric dimension ''d''. This is the well known [[isoperimetry|isoperimetric problem]] &mdash; as discussed above, for the Euclidean space the constant ''C'' is known precisely since the minimum is achieved for the ball. 

An infinite cylinder (i.e. a [[cartesian product|product]] of the [[unit circle|circle]] and the [[real line|line]]) 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 ''C''). Any compact manifold has isoperimetric dimension&nbsp;0.

It is also possible for the isoperimetric dimension to be larger than the topological dimension. The simplest example is the infinite [[jungle gym]], which has topological dimension 2 and isoperimetric dimension 3. See [https://web.archive.org/web/20040817075143/http://www.math.ucla.edu/~bon/jungle.html] for pictures and Mathematica code.

The [[hyperbolic geometry|hyperbolic plane]] has topological dimension 2 and isoperimetric dimension infinity. In fact the hyperbolic plane has positive [[Cheeger constant]]. This means that it satisfies the inequality

:<math>\operatorname{area}(\partial D)\geq C\operatorname{vol}(D),</math>

which obviously implies infinite isoperimetric dimension.

==Consequences of isoperimetry==<!-- This section is linked from [[Random walk]] -->

A simple integration over ''r'' (or sum in the case of graphs) shows that a ''d''-dimensional isoperimetric inequality implies a ''d''-dimensional [[Growth rate (group theory)|volume growth]], namely

:<math>\operatorname{vol} B(x,r)\geq Cr^d</math>

where ''B''(''x'',''r'') denotes the ball of radius ''r'' around the point ''x'' in the [[Riemannian manifold|Riemannian distance]] or in the [[Glossary of graph theory#Distance|graph distance]]. 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 '''Z''' (i.e. all the integers with edges between ''n'' and ''n''&nbsp;+&nbsp;1) and connecting to the vertex ''n'' a complete [[binary tree]] of height |''n''|. Both properties (exponential growth and 0 isoperimetric dimension) are easy to verify.

An interesting exception is the case of [[Group (mathematics)|groups]]. It turns out that a group with polynomial growth of order ''d'' has isoperimetric dimension ''d''. This holds both for the case of [[Lie group]]s and for the [[Cayley graph]] of a [[finitely generated group]].

A theorem of [[Nicholas Varopoulos|Varopoulos]] connects the isoperimetric dimension of a graph to the rate of escape of [[random walk]] on the graph. The result states

''Varopoulos' theorem: If G is a graph satisfying a d-dimensional isoperimetric inequality then''

:<math>p_n(x,y)\leq Cn^{-d/2} </math>

''where'' <math display="inline"> p_n(x,y)</math> ''is the probability that a random walk on'' ''G'' ''starting from'' ''x'' ''will be in'' ''y'' ''after'' ''n'' ''steps, and'' ''C'' ''is some constant.''

==References==
<references />
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* Isaac Chavel, ''Isoperimetric Inequalities: Differential geometric and analytic perspectives'', Cambridge university press, Cambridge, UK (2001), {{ISBN|0-521-80267-9}}
:Discusses the topic in the context of manifolds, no mention of graphs.
* N. Th. Varopoulos, ''Isoperimetric inequalities and Markov chains'', J. Funct. Anal. '''63:2''' (1985), 215–239.
* Thierry Coulhon and Laurent Saloff-Coste, ''Isopérimétrie pour les groupes et les variétés'', Rev. Mat. Iberoamericana '''9:2''' (1993), 293–314.
:This paper contains the result that on groups of polynomial growth, volume growth and isoperimetric inequalities are equivalent. In French.
* Fan Chung, ''Discrete Isoperimetric Inequalities''. ''Surveys in Differential Geometry IX'', International Press, (2004), 53–82. http://math.ucsd.edu/~fan/wp/iso.pdf.
:This paper contains a precise definition of the isoperimetric dimension of a graph, and establishes many of its properties.

[[Category:Mathematical analysis]]
[[Category:Dimension]]