Asymptotic Isoperimetry of Balls in Metric Measure Spaces

In this paper, we study the asymptotic behavior of the volume of spheres in metric measure spaces. We first introduce a general setting adapted to the study of asymptotic isoperimetry in a general class of metric measure spaces. Let A be a family of subsets of a metric measure space (X, d, μ), with finite, unbounded volume. For t > 0, we define: I ↓ A(t) = inf A∈A,μ(A)≥t μ(∂A). We say that A is asymptotically isoperimetric if ∀t > 0: I ↓ A(t) ≤ CI(t), where I is the profile of X. We show that there exist graphs with uniform polynomial growth whose balls are not asymptotically isoperimetric and we discuss the stability of related properties under quasi-isometries. Finally, we study the asymptotically isoperimetric properties of connected subsets in a metric measure space. In particular, we build graphs with uniform polynomial growth whose connected subsets are not asymptotically isoperimetric.