On \(L_p\)-Brunn-Minkowski type and \(L_p\)-isoperimetric type inequalities for general measures

• Published in: arXiv.org 2020-06
• Main Authors: Roysdon, Michael, Sudan Xing
• Format: Article
• Language: English
• Subjects: Borel sets; Inequalities; Inequality
• ISSN: 2331-8422

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.