A New Excluding Condition towards the Soprunov-Zvavitch conjecture on Bezout-type inequalities

2023/02/02 by Szusterman, Maud
#52A20 (Primary) #52B60 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2302.01213

Abstract

In 2015, I. Soprunov and A. Zvavitch have shown how to use the Bernstein-Khovanskii-Kushnirenko theorem to derive non-negativity of a certain bilinear form F_Δ, defined on (pairs of) convex bodies. Together with C. Saroglou, they proved non-negativity of F_K characterizes simplices, among all polytopes. It is conjectured the characterization further holds among all convex bodies. Towards this conjecture, several necessary conditions on K (for non-negativity of F_K), were derived. We give a new necessary condition, expressed with isoperimetric ratios, which provides a further step towards a (conjectural) characterization of simplices among a certain subclass of convex bodies.

Related

• Extemizers in Soprunov and Zvavitch's Bezout inequalities for mixed volumes
• Bézout type inequality in convex geometry
• General Bezout-type theorems
• Semigroups, Cartier divisors and convex bodies
• Shephard's inequalities, Hodge-Riemann relations, and a conjecture of Fedotov