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Asymptotic isoperimetry of balls in metric measure spaces. (English) Zbl 1116.53028

Let \((X, d, \mu)\) be a metric measure space, where \(d\) is a distance on \(X\), \(\mu\) is a \(\sigma\)-finite Borel measure on \(X\). For any subset \(A\) of \(X\) and any \(h>0\), the set \[ \partial_h A:=\{x\in X: d(x,A)\leq h, \, d(x,A^c)\leq h\} \] is called the \(h\)-boundary of \(A\). Let us call the \(h\)-profile the nondecreasing function defined on \(\mathbb R_+\) by
\[ I_h(t):=\inf_{\mu(A)\geq t} \mu(\partial_h A), \]
where \(A\) ranges over all \(\mu\)-measurable subsets of \(X\). Let \(\mathcal A\) be a family of subsets of \(X\) with finite, unbounded volumes. The author calls the nondecreasing function
\[ I^\downarrow_{h,{\mathcal A}}(t)=\inf_{\mu(A)\geq t, A\in{\mathcal A}} \mu(\partial_h A) \] lower \(h\)-profile restricted to \(\mathcal A\) The family \(\mathcal A\) is called asymptotically isoperimetric if there exist constants \(C_1\) and \(C_2\) such that \( I^\downarrow_{h,{\mathcal A}}(t)\leq C_1\, I_h(C_2t)\) for all \(t\in{\mathbb R}_+\). He studies asymptotically isoperimetric properties of balls in a metric measure spaces and discusses the stability of related properties under quasi-isometries. Also, the author studies the asymptotically isoperimetric properties of connected subsets in a metric measure space and builds graphs with uniform polynomial growth whose connected subsets are not asymptotically isoperimetric.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
49Q20 Variational problems in a geometric measure-theoretic setting
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