Concept

Isoperimetric dimension

Summary
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 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 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 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.
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