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

Abstract

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