  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.
