On the improvement of concavity of convex measures (Q2789875)

The main result of the paper is the following. Let \(\mu\) and \(A\) be an unconditional log-concave measure on \({\mathbb R}^n\) and an unconditional convex subset of \({\mathbb R}^n\), relatively. Then, for every \(A_{1}, A_{2} \in \{\alpha A; \alpha>0\}\) and for every \(\lambda\in [0; 1]\) NEWLINE\[NEWLINE \mu((1-\lambda)A_{1} +\lambda A_{2})^{1/n} \geq (1-\lambda)\mu(A_{1})^{1/n} +\lambda\mu(A_{2})^{1/n}. NEWLINE\]NEWLINE This is a generalization of the Brunn-Minkowski inequality.