November 30, 2025

On the volume of sums of anti-blocking convex bodies

AMAuttawich Manui CNCheikh Saliou NdiayeCentre National de la Recherche Scientifique AZArtem Zvavitch

Key Points

Volume inequalities show relationships in anti-blocking convex bodies, highlighting geometric structures and properties.
Key findings include Plünnecke–Ruzsa-type and Milman inequalities related to volume and projection ratios.
The approach is grounded in geometric analysis, applying techniques to develop inequalities and sharp constants.
Support for the generalized case could fulfill significant gaps in understanding densities and their implications for geometric structures.

Abstract

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.

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