On the volume of sums of anti-blocking convex bodies

In this paper, we study volume inequalities of Minkowski sum in the class of anti-blocking convex bodies. We prove analogues of a Plünnecke–Ruzsa-type inequality and Milman inequality on the concavity of the ratio of volumes of bodies and their projections. We also study Firey Formula: see text-sums of anti-blocking convex bodies, and prove a Plünnecke–Ruzsa-type inequality, Milman inequality and Rogers–Shephard inequality. The sharp constants are provided in all of those inequalities, for the class of anti-blocking convex bodies. Finally, we extend our results to the case of unconditional product measures with non-increasing density.