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

Arnaud Marsiglietti (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We consider a different $L^p$-Minkowski combination of compact sets in $\mathbb{R}^n$ than the one introduced by Firey and we prove an $L^p$-Brunn-Minkowski inequality, $p \in [0,1]$, 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 $t \mapsto \mu(t^{\frac{1}{p}} A)$, $p \in (0,1]$, for unconditional convex measures $\mu$ and unconditional convex body $A$ in $\mathbb{R}^n$. 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.

Mots clés: Brunn-Minkowski-Firey theory; $L^p$-Minkowski combination; convex body; convex measure; (B)-conjecture

Domaines: Analyse fonctionnelle [math.FA]

Arnaud Marsiglietti. A note on an $L^p$-Brunn-Minkowski inequality for convex measures in the unconditional case. 2014. ⟨hal-01081764⟩