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The Extremals of the Alexandrov-Fenchel Inequality for Convex Polytopes

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.