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Complemented Brunn-Minkowski Inequalities and Isoperimetry for Homogeneous and Non-Homogeneous Measures

Elementary proofs of sharp isoperimetric inequalities on a normed space $(\mathbb{R}^n,||\cdot||)$ equipped with a measure $μ= w(x) dx$ so that $w^p$ is homogeneous are provided, along with a characterization of the corresponding equality cases. When $p \in (0,\infty]$ and in addition $w^p$ is assumed concave, the result is an immediate corollary of the Borell-Brascamp-Lieb extension of the classical Brunn-Minkowski inequality, providing an elementary proof of a recent result of Cabré-Ros Oton-Serra. When $p \in (-1/n,0)$, the relevant property turns out to be a novel "complemented Brunn-Minkowski" inequality, which we show is always satisfied by $μ$ when $w^p$ is homogeneous. This gives rise to a new class of measures, which are "complemented" analogues of the class of convex measures introduced by Borell, but which have vastly different properties. The resulting isoperimetric inequality and characterization of isoperimetric minimizers extends beyond the recent results of Cañete--Rosales and Howe. The isoperimetric and Brunn-Minkowski type inequalities extend to the non-homogeneous setting, under a certain log-convexity assumption on the density. Finally, we obtain functional, Sobolev and Nash-type versions of the studied inequalities.