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

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.