Shephard's inequalities, Hodge-Riemann relations, and a conjecture of Fedotov

TL;DR

This paper disproves Fedotov's conjecture on higher-order Shephard inequalities by demonstrating its contradiction with Hodge-Riemann relations for convex polytopes, providing insights into geometric inequalities.

Contribution

It provides a counterexample to Fedotov's conjecture, linking geometric inequalities with Hodge-Riemann relations and clarifying their algebraic and geometric aspects.

Findings

Fedotov's conjecture is false due to contradiction with Hodge-Riemann relations.

Disproved higher-order Shephard inequalities for convex bodies.

Provides expository insights on the algebraic and geometric nature of these inequalities.

Abstract

A well-known family of determinantal inequalities for mixed volumes of convex bodies were derived by Shephard from the Alexandrov-Fenchel inequality. The classic monograph Geometric Inequalities by Burago and Zalgaller states a conjecture on the validity of higher-order analogues of Shephard's inequalities, which is attributed to Fedotov. In this note we disprove Fedotov's conjecture by showing that it contradicts the Hodge-Riemann relations for simple convex polytopes. Along the way, we make some expository remarks on the linear algebraic and geometric aspects of these inequalities.