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. The
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 :\operatorname{area}(\partial D)\geq C\operatorname{vol}(D), which obviously implies infinite isoperimetric dimension. ==Consequences of isoperimetry== 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 :\operatorname{vol} B(x,r)\geq Cr^d 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. A theorem of
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'' :p_n(x,y)\leq Cn^{-d/2}
where p_n(x,y)
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==