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 $\mathcal{A}$ be a family of subsets of a metric measure space $(X,d,\mu)$, with finite, unbounded volume. For $t > 0$, we define
$$
I^{\downarrow}_{\mathcal{A}}(t)=\inf_{A\in \mathcal{A},\, \mu(A)\geq t} \mu(\partial A).
$$
We say that $\mathcal{A}$ is asymptotically isoperimetric if $\forall\; t>0$
$$
I_{\mathcal{A}}^{\downarrow}(t)\leq CI(Ct),
$$
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.