| Abstrakt: |
In 2011 Lutwak, Yang and Zhang extended the definition of the Lp-Minkowski convex combination (p ≥ 1) introduced by Firey in the 1960s from convex bodies containing the origin in their interiors to all measurable subsets in Rn, and as a consequence, extended the Lp-Brunn-Minkowski inequality (Lp-BMI) to the setting of all measurable sets. In this paper, we present a functional extension of their Lp-Minkowski convex combination—the Lp,s–supremal convolution and prove the Lp-Borell-Brascamp-Lieb type (Lp-BBL) inequalities. Based on the Lp-BBL type inequalities for functions, we extend the Lp-BMI for measurable sets to the class of Borel measures on Rn having (1/s)-concave densities, with s ≥ 0; that is, we show that, for any pair of Borel sets A,B ⊂ Rn, any t ∈ [0,1] and p ≥ 1, one has μ((1-t)⋅p A +p t ⋅p B)p/n+s ≥ (1-t) μ (A)p/n+s + t μ (B)p/n+s, where μ is a measure on Rn having a (1/s)-concave density for 0 ≤ s < ∞. Additionally, with the new defined Lp,s–supremal convolution for functions, we prove Lp-BMI for product measures with quasi-concave densities and for log-concave densities, Lp-Prékopa-Leindler type inequality (Lp-PLI) for product measures with quasi-concave densities, Lp-Minkowski's first inequality (Lp-MFI) and Lp isoperimetric inequalities (Lp-ISMI) for general measures, etc. Finally a functional counterpart of the Gardner-Zvavitch conjecture is presented for the p-generalization. [ABSTRACT FROM AUTHOR] |
