The extremals of the Alexandrov–Fenchel inequality for convex polytopes

Yair Shenfeld, Ramon van Handel

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

The Alexandrov–Fenchel inequality, a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes, lies at the heart of convex geometry. The characterization of its extremal bodies is a long-standing open problem that dates back to Alexandrov’s original 1937 paper. The known extremals already form a very rich family, and even the fundamental conjectures on their general structure, due to Schneider, are incomplete. In this paper, we completely settle the extremals of the Alexandrov–Fenchel inequality for convex polytopes. In particular, we show that the extremals arise from the combination of three distinct mechanisms: translation, support, and dimensionality. The characterization of these mechanisms requires the development of a diverse range of techniques that shed new light on the geometry of mixed volumes of non-smooth convex bodies. Our main result extends further beyond polytopes in a number of ways, including to the setting of quermassintegrals of arbitrary convex bodies. As an application of our main result, we settle a question of Stanley on the extremal behavior of certain log-concave sequences that arise in the combinatorics of partially ordered sets.

Original languageEnglish (US)
Pages (from-to)89-204
Number of pages116
JournalActa Mathematica
Volume231
Issue number1
DOIs
StatePublished - 2023

All Science Journal Classification (ASJC) codes

  • General Mathematics

Keywords

  • Alexandrov–Fenchel inequality
  • convex polytopes
  • extremum problems in geometry and combinatorics
  • mixed volumes

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