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 Rⁿ than the one introduced by Firey and we prove an L ^p-Brunn-Minkowski inequality, p ∈ [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 ↦ μ(t^1/pA), p ∈ (0,1), for unconditional convex measures μ and unconditional convex body A in Rⁿ. 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.