A relation between Mahler volume and flag number for convex polytopes

Abstract

Given a convex polytope $P\subset {\mathbb{R}}^d$ with $\mathcal{F}(P)$ many flags, we prove $$\mathrm{vol}(P)\mathrm{vol}(P-P{)}^{\circ }\le \frac{\mathcal{F}(P)}{(d!{)}^2}.$$ This implies the following relation between Mahler volume and the number of flag conjectured by Freij, Schmitt, Schymura and Ziegler: for a centrally symmetric polytope $P\subset {\mathbb{R}}^d$ holds $$\mathrm{vol}(P)\mathrm{vol}(P^{\circ })\le \frac{2^d}{(d!{)}^2}\mathcal{F}(P).$$ This shows that the Mahler conjecture implies Kalai's flag conjecture.