On $L_{p}$ Brunn–Minkowski type inequalities for a general class of functionals

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} ((1-\lambda)\cdot K+_{p} \lambda \cdot L )^{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.