Көпөлшемді кеңістіктердің изопериметриялық өлшемі

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 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.

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

which obviously implies infinite isoperimetric dimension.

The isoperimetric dimension of graphs can be defined in a similar fashion. A precise definition is given in Chung's survey. Area and volume are measured by set sizes. 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

(This MathOverflow question provides more details.) The graph analogs of all the examples above hold but the definition is slightly different in order to avoid that the isoperimetric dimension of any finite graph is 0: In the above formula the volume of is replaced by (see Chung's survey, section 7). 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 ∞.