In the Euclidean space, the 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.
We say about a manifold M that it satisfies a d-dimensional isoperimetric inequality if for any open set D in M with a smooth boundary one has
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 − 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 supremum of all values of d such that M satisfies a d-dimensional isoperimetric inequality.
A d-dimensional Euclidean space has isoperimetric dimension d. This is the well known isoperimetric problem — 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 product of the circle and the 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 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 for pictures and Mathematica code.
which obviously implies infinite isoperimetric dimension.
The isoperimetric dimension of graphs can be defined in a similar fashion. there is no need to have an area and volume measures. One simply counts points. For every subset A of the graph G one defines as the set of vertices in with a neighbor in A. A d-dimensional isoperimetric inequality is now defined by
The graph analogs of all the examples above hold. The isoperimetric dimension of any finite graph is 0. The isoperimetric dimension of a d-dimensional grid is d. In general, the isoperimetric dimension is preserved by quasi isometries, both by quasi-isometries between manifolds, between graphs, and even by quasi isometries carrying manifolds to graphs, with the respective definitions. In rough terms, this means that a graph "mimicking" a given manifold (as the grid mimics the Euclidean space) would have the same isoperimetric dimension as the manifold. An infinite complete binary tree has isoperimetric dimension ∞.
A simple integration over r (or sum in the case of graphs) shows that a d-dimensional isoperimetric inequality implies a d-dimensional volume growth, namely
where B(x,r) denotes the ball of radius r around the point x in the Riemannian distance or in the 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 + 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 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 groups and for the Cayley graph of a finitely generated group.
Varopoulos' theorem: If G is a graph satisfying a d-dimensional isoperimetric inequality then
where is the probability that a random walk on G starting from x will be in y after n steps, and C is some constant.