  \footnotesize B\"{o}r\"{o}czky, Lutwak, Yang and Zhang recently conjectured a
certain strengthening of the Brunn-Minkowski inequality for symmetric convex
bodies, the so-called log-Brunn-Minkowski inequality. We establish this
inequality together with its equality cases for pairs of unconditional convex
bodies with respect to some orthonormal basis. Applications of this fact are
discussed. Moreover, we prove that the log-Brunn-Minkowski inequality is
equivalent to the (B)-Theorem for the uniform measure of the cube (this has
been proven by Cordero-Erasquin, Fradelizi and Maurey for the gaussian measure
instead).
