On p-Brunn-Minkowski and Brascamp-Lieb inequalities

We show that a strong version of the Brascamp–Lieb inequality for symmetric log-concave measure with $\alpha$-homogeneous potential $V$ is equivalent to a $p$-Brunn–Minkowski inequality for level sets of $V$ with some $p(\alpha ,n)<0$. We establish links between several inequalities of this type on the sphere and the Euclidean space. Exploiting these observations, we prove new sufficient conditions for symmetric $p$-Brunn–Minkowski inequality with $p<1$. In particular, we prove the local log-Brunn–Minkowski for $L_q$-balls for all $q\ge 1$ in all dimensions, which was previously known only for $q\ge 2$.