  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.
We then introduce a notion of "being asymptotically isoperimetric" for a family
of finite and unbounded measure subsets. 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.
