On $L_p$ Brunn–Minkowski type inequalities for a general class of functionals

Abstract

The $L_p$ version (for $p>1$) of the dimensional Brunn-Minkowski inequality for the standard Gaussian measure ${\gamma }_n(\cdot )$ on ${\mathbb{R}}^n$ is shown. More precisely, we prove that for any $0$-symmetric convex sets with nonempty interior, any $p>1$, and every $\lambda \in (0,1)$,

with equality, for some $\lambda \in (0,1)$ and $p>1$, if and only if $K=L$. This result, recently established without the equality conditions by Hosle, Kolesnikov and Livshyts, by using a different and functional approach, turns out to be the $L_p$ extension of a celebrated result for the Minkowski sum (that is, for $p=1$) by Eskenazis and Moschidis (2021) on a problem by Gardner and Zvavitch (2010). Moreover, an $L_p$ Brunn–Minkowski type inequality is obtained for the classical Wills functional $\mathcal{W}(\cdot )$ of convex bodies. These results are derived as a consequence of a more general approach, which provides us with other remarkable examples of functionals satisfying $L_p$ Brunn–Minkowski type inequalities, such as different absolutely continuous measures with radially decreasing densities.