On $p$-Brunn-Minkowski inequalities for intrinsic volumes with $0\le p<1$

Abstract

We prove the validity of the $p$-Brunn-Minkowski inequality for the intrinsic volume $V_k$, $k=2,\dots ,n-1$, of convex bodies in ${\mathbb{R}}^n$, in a neighborhood of the unit ball, for $0\le p<1$. We also prove that this inequality does not hold true on the entire class of convex bodies of ${\mathbb{R}}^n$, when $p$ is sufficiently close to $0$.