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

Abstract

In this work, 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 $\lambda \in (0,1)$, \[ \gamma_n\bigl((1-\lambda)\cdot K+_p \lambda \cdot L\bigr)^{p/n} \geqslant (1-\lambda ) \gamma_n(K)^{p/n} + \lambda \gamma_n(L)^{p/n}, \] 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.