Summary: 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, \(L_p\)-Prékopa-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.