A nonabelian Brunn-Minkowski inequality II

We prove that every unimodular locally compact group $G$ of noncompact Lie dimension $n$ satisfies the sharp Brunn–Minkowski inequality $${\mu }_G(XY{)}^{1\mathrm{/}n}\ge {\mu }_G(X{)}^{1\mathrm{/}n}+{\mu }_G(Y{)}^{1\mathrm{/}n},$$ 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.