  In this paper we consider nonnegatively curved finite dimensional Alexandrov
spaces such that unit balls have volumes uniformly bounded from below away from
zero. We study the relation between the isoperimetric profile, the existence of
isoperimetric sets, and the asymptotic structure at infinity of such spaces.
  In the previous setting, we prove that the following conditions are
equivalent: the space has linear volume growth; it is Gromov--Hausdorff
asymptotic to one cylinder at infinity; it has uniformly bounded isoperimetric
profile; the entire space is a tubular neighborhood of either a line or a ray.
Moreover, on a space satisfying any of the previous conditions, we prove
existence of isoperimetric sets for sufficiently large volumes, and, in such
generality, we characterize the geometric rigidity at the level of the
isoperimetric profile.
  Specializing our study to the $2$-dimensional case, we prove that unit balls
have always volumes uniformly bounded from below away from zero, and we prove
existence of isoperimetric sets for every volume, characterizing also their
topology when the space has no boundary.
  The proofs exploit an approach by direct method and the results in particular
apply to Riemannian manifolds with nonnegative sectional curvature and to
Euclidean convex bodies. Up to the authors' knowledge, most of the results are
new even in these smooth cases.
