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arXiv · 2109.05169

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

Ramon van Handel
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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.

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Ramon van Handel. 2021-09-11. Shephard's inequalities, Hodge-Riemann relations, and a conjecture of Fedotov. https://arxiv.org/abs/2109.05169

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