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

TL;DR

This paper introduces a new necessary condition involving isoperimetric ratios that advances the understanding of the Soprunov-Zvavitch conjecture related to Bezout-type inequalities and convex bodies.

Contribution

It provides a novel necessary condition based on isoperimetric ratios, moving closer to characterizing simplices among convex bodies regarding the non-negativity of a bilinear form.

Findings

New necessary condition involving isoperimetric ratios.

Progress towards characterizing simplices among convex bodies.

Enhanced understanding of the Soprunov-Zvavitch conjecture.

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 $F_{\Delta}$, 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.