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

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 $0le p<1$. 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 $0$.