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 μ(∂A). A∈A, μ(A)≥t We say that A is asymptotically isoperimetric if ∀ t > 0 I↓_A(t) ≤ CI(Ct), 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.