Cone-volume measures of polytopes

Abstract

The cone-volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition. As a consequence a conjectured (a dozen years ago) fundamental sharp affine isoperimetric inequality for the U-functional is completely established -along with its equality conditions. 2010 Mathematics Subject Classification. 52A40, 52B11.

Cone-volume measure of general centered convex bodies

“…In fact, in [23] it was shown by M. Henk and E. Linke that the necessity part of Theorem I also holds for centered polytopes, i.e., Theorem II. (See [23,Theorem 1.1].) Let P ∈ K n be a centered polytope.…”

Affine images of isotropic measures

“…Added in proof: After this paper was submitted for publication, an affirmative answer to Problem 8.9, for polytopes, was given by Henk and Linke [31].…”

Necessary subspace concentration conditions for the even dual Minkowski problem

“…An extension of the validity of inequality (1.8) to centered bodies, i.e., bodies whose center of mass is at the origin, was given in the discrete case by Henk and Linke [29], and in the general setting by Böröczky and Henk [7]. For a related stability result concerning (1.8) we refer to [8].…”

“…The cone-volume measure for convex bodies has been studied extensively over the last few years in many different contexts, see, e.g., [4,5,10,11,12,23,25,27,28,29,30,31,34,35,39,40,41,42,43,44,47,51,55,56]. One very important property of the cone-volume measure -and which makes it so essential -is its SL(n)-invariance, or simply called affine invariance.…”