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.
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|.
The result states Varopoulos' theorem: If G is a graph satisfying a d-dimensional isoperimetric inequality then where