On $L_p$-Brunn-Minkowski type and $L_p$-isoperimetric type inequalities for general measures

In 2011 Lutwak, Yang and Zhang extended the definition of the $L_p$-Minkowski convex combination ($p \geq 1$) introduced by Firey in the 1960s from convex bodies containing the origin in their interiors to all measurable subsets in $\mathbb{R}^n$, and as a consequence, extended the $L_p$-Brunn-Minkowski inequality ($L_p$-BMI) to the setting of all measurable sets. In this paper, we present a functional extension of their $L_p$-Minkowski convex combination---the $L_{p,s}$--supremal convolution and prove the $L_p$-Borell-Brascamp-Lieb type ($L_p$-BBL) inequalities. Based on the $L_p$-BBL type inequalities for functions, we extend the $L_p$-BMI for measurable sets to the class of Borel measures on $\mathbb{R}^n$ having $\left(\frac{1}{s}\right)$-concave densities, with $s \geq 0$; that is, we show that, for any pair of Borel sets $A,B \subset \mathbb{R}^n$, any $t \in [0,1]$ and $p\geq 1$, one has $$\mu((1-t) \cdot_p A +_p t \cdot_p B)^{\frac{p}{n+s}} \geq (1-t) \mu(A)^{\frac{p}{n+s}} + t \mu(B)^{\frac{p}{n+s}},$$ where $\mu$ is a measure on $\mathbb{R}^n$ having a $\left(\frac{1}{s}\right)$-concave density for $0 \leq s < \infty$. Additionally, with the new defined $L_{p,s}$--supremal convolution for functions, we prove $L_p$-BMI for product measures with quasi-concave densities and for log-concave densities, $L_p$-Pr\'ekopa-Leindler type inequality ($L_p$-PLI) for product measures with quasi-concave densities, $L_p$-Minkowski's first inequality ($L_p$-MFI) and $L_p$ isoperimetric inequalities ($L_p$-ISMI) for general measures, etc. Finally a functional counterpart of the Gardner-Zvavitch conjecture is presented for the $p$-generalization.