Isoperimetric Inequalities in A Metric Measure Space
Most of the work on isoperimetric problem has been done in the context of smooth regions in Euclidean spaces, or more generally, in Riemannian manifolds. However, the isoperimetric problem can be formulated in much greater generality, using the notion of Minkowski content. Let be a metric measure space: X is a metric space with metric d, and μ is a Borel measure on X. The boundary measure, or Minkowski content, of a measurable subset A of X is defined as the lim inf
where
is the ε-extension of A.
The isoperimetric problem in X asks how small can be for a given μ(A). If X is the Euclidean plane with the usual distance and the Lebesgue measure then this question generalizes the classical isoperimetric problem to planar regions whose boundary is not necessarily smooth, although the answer turns out to be the same.
The function
is called the isoperimetric profile of the metric measure space . Isoperimetric profiles have been studied for Cayley graphs of discrete groups and for special classes of Riemannian manifolds (where usually only regions A with regular boundary are considered).
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