A nonabelian Brunn-Minkowski inequality II

Abstract

We prove that every unimodular locally compact group $G$ of noncompact Lie dimension $n$ satisfies the sharp Brunn--Minkowski inequality \[ \mu_G(XY)^{1/n}\ge\mu_G(X)^{1/n}+\mu_G(Y)^{1/n}, \] and establish a general form for arbitrary, possibly nonunimodular, locally compact groups. This fully confirms the nonabelian Brunn--Minkowski conjecture proposed by the present authors and Zhang. As an application, we obtain an isoperimetric inequality on symmetric spaces of noncompact type.