Acta Mathematica

Volume 231 (2023)

Number 1

The extremals of the Alexandrov–Fenchel inequality for convex polytopes

Pages: 89 – 204



Yair Shenfeld (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.; and Division of Applied Mathematics, Brown University, Providence, Rhode Island, U.S.A.)

Ramon van Handel (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)


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.


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

2010 Mathematics Subject Classification

05B25, 52A39, 52A40, 52B05

This work was supported in part by NSF grant DMS-1811735, and by the Simons Collaboration on Algorithms & Geometry.

Received 10 November 2020

Accepted 30 January 2022

Published 29 September 2023