A note on an $L^p$-Brunn–Minkowski inequality for convex measures in the unconditional case

Arnaud Marsiglietti

Vol. 277 (2015), No. 1, 187–200

DOI: 10.2140/pjm.2015.277.187

Abstract

We consider a different $L^p$-Minkowski combination of compact sets in ${ℝ}^n$ than the one introduced by Firey and we prove an $L^p$-Brunn–Minkowski inequality, $p\in \left[\mathfrak{0},\mathfrak{1}\right]$, for a general class of measures called convex measures that includes log-concave measures, under unconditional assumptions. As a consequence, we derive concavity properties of the function $t\mapsto \mu \left(t^{\mathfrak{1}∕p}A\right)$, $p\in \left(\mathfrak{0},\mathfrak{1}\right]$, for unconditional convex measures $\mu$ and unconditional convex body $A$ in ${ℝ}^n$. We also prove that the (B)-conjecture for all uniform measures is equivalent to the (B)-conjecture for all log-concave measures, completing recent works by Saroglou.

Keywords

Brunn–Minkowski–Firey theory, $L^p$-Minkowski combination, convex body, convex measure, (B)-conjecture

Mathematical Subject Classification 2010

Primary: 28A75, 52A40, 60B11

Milestones

Received: 25 November 2014

Revised: 13 March 2015

Accepted: 14 March 2015

Published: 6 August 2015