The extremals of the Alexandrov–Fenchel inequality for convex polytopes

Abstract

extremals of the alexandrov-fenchel inequality 95 of the settings considered by Stanley, where N i is the number of linear extensions of a partially ordered set for which a distinguished element has rank i.Such extremal problems appear to be inaccessible by currently known methods of enumerative or algebraic combinatorics.This example highlights the significance of the questions considered in this paper to extremal problems in other areas of mathematics, and hints at the possibility that the structures developed here might have analogues outside convexity; a brief discussion of algebraic analogues of our results is given in §16.Let us note that, far from being esoteric, it is precisely the case of convex bodies with empty interior (which is not covered by previous conjectures) that arises in combinatorial applications [33].This reinforces the importance of a complete characterization of the extremals, whose formulation we turn to presently. Three extremal mechanismsThe aim of this section is to formulate and explain the main result of this paper.We first recall some key facts on mixed volumes and mixed area measures.We will subsequently describe three distinct mechanisms that give rise to extremals of the Alexandrov-Fenchel inequality, and state our main result.Here and throughout the paper, our standard reference on convexity is the monograph [30]. Basic facts2.1.1.Convex bodies, mixed volumes, mixed area measures Fix n⩾3.A convex body is a non-empty compact convex set in R n .A (convex) polytope is the convex hull of a finite number of points.With each convex body K, we associate its support functionWe think of h K either as a function on S n-1 or as a 1-homogeneous function on R n .Geometrically, if u∈S n-1 , then h K (u) is the (signed) distance to the origin of the supporting hyperplane of K with outer normal u; thus h K : S n-1 !R uniquely determines K, as any convex body is the intersection of its supporting half-spaces.The key property of support functions is that they behave naturally under addition, that is,

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