  The sharp isoperimetric inequality for non-compact Riemannian manifolds with
non-negative Ricci curvature and Euclidean volume growth has been obtained in
increasing generality with different approaches in a number of contributions
[arXiv:1812.05022, arXiv:2012.09490, arXiv:2009.13717, arXiv:2103.08496]
culminated by Balogh and Kristaly [arXiv:2012.11862] covering also m.m.s.'s
verifying the non-negative Ricci curvature condition in the synthetic sense of
Lott, Sturm and Villani. In sharp contrast with the compact case of positive
Ricci curvature, for a large class of spaces including weighted Riemannian
manifolds, no complete characterisation of the equality cases is present in the
literature.
  The scope of this note is to settle this problem by proving, in the same
generality of [arXiv:2012.11862], that the equality in the isoperimetric
inequality can be attained only by metric balls. Whenever this happens the
space is forced, in a measure theoretic sense, to be a cone.
  Our result applies to different frameworks yielding as corollaries new
rigidity results: it extend to weighted Riemannian manifold the rigidity
results of [arXiv:2009.13717], it extend to general $\mathsf{RCD}$ spaces the
rigidity results of [arXiv:2201.04916] and finally applies also to the
Euclidean setting by proving that that optimisers in the anisotropic and
weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff
shapes.
