A concavity property for the measure of product sets in groups

Let G be a connected locally compact group with a left invariant Haar measure μ. We prove that the function ξ(x) = inf {μ̅(AB): μ(A) = x} is concave for any fixed bounded set B ⊂ G. This is used to give a new proof of Kemperman's inequality ${\displaystyle \mu ̲\left(AB\right)\ge min(\mu ̲\left(A\right)+\mu ̲\left(B\right),\mu \left(G\right))}$ for unimodular G.