Cone-volume measures of polytopes

By Martin Henk, Eva Linke. 2013

104 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Pharmacology

View Paper: https://www.semanticscholar.org/paper/72ec25a7fe15ed2b7742464bee0c795f44d0f684

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In Applied Mathematics

In Pharmacology

Citations in the corpus

Dar’s conjecture and the log–Brunn–Minkowski inequality (2016, background): “Recently, Henk and Linke [28] proved that polytopes in R with centroid at o satisfy the subspace concentration condition.”
On the L dual Minkowski problem (2018, background): “See, for example, Barthe, Guédon, Mendelson & Naor [4], Böröczky & Henk [7], Böröczky, Lutwak, Yang & Zhang [9–11], Henk & Linke [29], Ludwig [44], Ludwig & Reitzner [47], Naor [54], Naor & Romik [55], Paouris & Werner [56], Stancu [62,63], Xiong [65], Zhu [70], and Zou & Xiong [73].”
Necessary subspace concentration conditions for the even dual Minkowski problem (2017, methodology): “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].”
Affine images of isotropic measures (2015, methodology): “9, for polytopes, was given by Henk and Linke [31].”
The discrete logarithmic Minkowski problem (2014, unknown): “oncentration condition. For more references on the progress of the subspace concentration condition, see, e.g., Henk et al. [32], He et al. [30], Xiong [67], B¨oro¨czky et al. [8], and Henk and Linke [31]. In [6], B¨oro¨czky, et al. established the following necessary and suﬃcient conditions for the”
Existence of solutions to the even dual Minkowski problem (2018, background): “It is still being intensively studied, see, for example, Barthe, Guédon, Mendelson & Naor [4], Böröczky & Henk [6], Böröczky-LYZ [8, 9, 10], Henk & Linke [31], Ludwig [38], Ludwig & Reitzner [40],”
Remarks on the conjectured log-Brunn–Minkowski inequality (2013, background): “entary to ξ′, for which euality holds in (i). It turns out that the subspace concentration condition arises naturally in the study of several other important problems in convex geometry (see e.g. [4] [7]). What still remains unknown concerning the logarithmic Minkowski problem, is the characterizati”
Cone-volume measure of general centered convex bodies (2016, background): “The next identity has been proved in the special case of polytopes in [23].”
The Minkowski problem in Gaussian probability space (2020, background): “Due to limit of space, we mention only a selection of beautiful results in this area, [10,12,33,34,54,56,60,61,63,65, 69,70,72] and refer the interested readers to Schneider’s book [66] for more details.”