On p-Brunn–Minkowski inequalities for intrinsic volumes, with 0le p

Abstract

We prove the validity of the p-Brunn–Minkowski inequality for the intrinsic volume V_k, k=2,dots , n-1, of symmetric convex bodies in {{mathbb {R}}}^n, in a neighbourhood of the unit ball when one of the bodies is the unit ball, for 0le 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.