Shephard’s Inequalities, Hodge-Riemann Relations, and a Conjecture of Fedotov

Research output: Chapter in Book/Report/Conference proceedingChapter

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.

Original languageEnglish (US)
Title of host publicationLecture Notes in Mathematics
PublisherSpringer Science and Business Media Deutschland GmbH
Pages337-354
Number of pages18
DOIs
StatePublished - 2023

Publication series

NameLecture Notes in Mathematics
Volume2327
ISSN (Print)0075-8434
ISSN (Electronic)1617-9692

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Keywords

  • Alexandrov-Fenchel inequality
  • Hodge-Riemann relations for convex polytopes
  • Mixed volumes
  • Shephard’s inequalities

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