The logarithmic Minkowski problem

Abstract

Abstract. In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

Affine images of isotropic measures

“…The following lemma was proved in [8]. For polytopes, the inequalitypart of the subspace concentration condition of Lemma 8.3 was established by He, Leng, and Li [30], with a shorter proof provided by Xiong [77].…”

“…In [8], the authors defined the subspace concentration condition of measures (defined below), which limits how concentrated a measure can be in a subspace. (This condition is connected with fully nonlinear partial differential equations.)…”

“…It was shown in [8] that the affine image of the cone-volume measure of a convex body is the cone-volume measure of the affine image of the body; i.e., if K is a convex body in R n that contains the origin in its interior, and A ∈ SL(n), then…”

The Lp Minkowski problem for polytopes for p<0

“…Establishing existence and uniqueness for the solution of the classical Minkowski problem was done by Aleksandrov, and Fenchel and Jessen (see, e.g., [52]). When p = 1, the L p Minkowski problem has been studied by, e.g., Lutwak [38], Lutwak and Oliker [39], Guan and Lin [18], Chou and Wang [10], Hug, et al [30], Böröczky, et al [5]. Additional references regarding the L p Minkowski problem and Minkowski-type problems can be found in [5, 8, 10, 17-21, 28-30, 32-34, 38, 39, 44, 53, 54].…”

“…In R n , necessary and sufficient conditions for the existence of the solution of the even L p Minkowski problem for the case of 0 < p < 1 was given by Haberl, et al [21]. Necessary and sufficient conditions for the existence of solutions to the even L 0 Minkowski problem (also called the logarithmic Minkowski problem) was recently established by Böröczky, et al [5]. Without the assumption that the measure is even, existence of solution of the PDE (1.1) for the case where p > −n were given by Chou and Wang [10].…”