The Extremals of the Alexandrov-Fenchel Inequality for Convex Polytopes

TL;DR

This paper fully characterizes the extremal bodies for the Alexandrov-Fenchel inequality in convex polytopes, revealing the mechanisms involved and extending the results to broader convex bodies and combinatorial sequences.

Contribution

It provides a complete description of extremals for the inequality in polytopes and introduces new techniques for analyzing mixed volumes of nonsmooth convex bodies.

Findings

Extremals arise from translation, support, and dimensionality mechanisms.

The characterization extends to quermassintegrals of general convex bodies.

Application to extremal behavior of log-concave sequences in combinatorics.

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 nonsmooth 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.