Relative entropy of cone measures and Lp centroid bodies
By Grigoris Paouris, Elisabeth M. Werner. 2009
165 citations recorded by the source corpus.
Let K be a convex body in ℝn. We introduce a new affine invariant, which we call ΩK, that can be found in three different ways: as a limit of normalized Lp‐affine surface areas; as the relative entropy of the cone measure of K and the cone measure of K°; as the limit of the volume difference of K and Lp‐centroid bodies. We investigate properties of ΩK and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show an ‘information inequality’ for convex bodies.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/a1d16402f59d4f07af7d9085e98a51063a5b9801
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In Applied Mathematics
- Yiming Zhao
- Yong Huang
- Deane Yang
- Gaoyong Zhang
- Erwin Lutwak
- Deping Ye
- Elisabeth M. Werner
- Qi-Rui Li
In Geometry and Topology
- Yiming Zhao
- Yong Huang
- Deane Yang
- Gaoyong Zhang
- Erwin Lutwak
- Elisabeth M. Werner
- Deping Ye
- Monika Ludwig
Citations in the corpus
- Cone-volume measure of general centered convex bodies (2016, unknown): “Werner [36], A.”
- Shadow systems of asymmetric Lp zonotopes (2013, unknown): “[9,12,21,23,26–30,32,35–37,44,46–48,51]).”
- Divergence for s-concave and log concave functions (2013, background): “Applications are given in the theory of convex bodies.”
- The logarithmic Minkowski problem (2012, unknown): “In recent years, cone-volume measures have appeared in e.g. [4], [19], [37], [38], [50], [51], [54], and [59].”
- Anisotropic fractional Sobolev norms (2013, unknown): “[10, 15–17,21– 25,29–32]).”
- Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems (2016, background): “The SL(n) invariance of the cone-volume measure makes the measure a useful notion in the geometry of normed spaces; see [10], [60], [61], [65], [66], [69].”
- Existence of solutions to the even dual Minkowski problem (2018, background): “Naor [47], Naor & Romik [48], Paouris & Werner [51], Stancu [61, 62], Xiong [65], Zhu [70], and Zou & Xiong [74].”
- Cone-volume measures of polytopes (2013, unknown): “, [2, 4, 5, 9, 16, 17, 20, 21, 22, 28].”