The discrete logarithmic Minkowski problem

By K. Boroczky, P. Hegedus, Guangxian Zhu. 2014

138 citations recorded by the source corpus.

Existence of solution of the logarithmic Minkowski problem is proved for the case where the discrete measures on the unit sphere satisfy the subspace concentration condition with respect to some special proper subspaces. In order to understand how optimal this condition is, we discuss certain conditions that any cone volume measure satisfies.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d6645d5ab485d8dec6a5f4ddbca06fc4d02bf9ab

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Yiming Zhao
2. Qi-Rui Li
3. Yong Huang
4. Shibing Chen
5. Jian Lu
6. Yan-nan Liu
7. Deping Ye
8. Dongmeng Xi

In Geometry and Topology

1. Yiming Zhao
2. Deping Ye
3. Qi-Rui Li
4. Shibing Chen
5. Sudan Xing
6. Yong Huang
7. Dongmeng Xi
8. Martin Henk

Citations in the corpus

• The Minkowski problem in Gaussian probability space (2020, background): “In the general case (non-even case), important contributions were made by Zhu [76], and later by Böröczky-Hegedűs-Zhu [9].”
• On the L dual Minkowski problem (2018, background): “See also Jian, Lu & Zhu [37], Lu & Wang [41], Zhu [71], etc., on this problem.”
• On the planar dual Minkowski problem (2018, background): “The L0-Minkowski problem, also called the logarithmic Minkowski problem [7, 10, 53], is highly interesting.”
• The Dual Orlicz–Minkowski Problem (2017, methodology): “inkowski problem attracted a lot of attention and amazing progress has been made (see e.g., [5, 6, 18, 21, 22, 33, 45, 61, 62, 63]). When p= 0, it becomes the logarithmic Minkowski problem (see e.g., [2, 4, 42, 43, 44, 60]). Note that solutions to the Lp Minkowski problem are the key ingredients in ”
• The p-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems (2017, methodology): “There are versions of Minkowski problems related to other functions, for instance, the L0 Minkowski or logarithmic Minkowski problems [5, 7, 38, 39, 40, 49] and the Orlicz-Minkowski problem [18, 22].”
• A flow method for the dual Orlicz–Minkowski problem (2020, background): “[5, 7, 8, 15, 17, 28, 32, 36, 39, 40, 41, 47, 48, 49, 50, 52, 54, 67, 68] and Schneider’s book [59], and corresponding references therein.”
• Cone-volume measure of general centered convex bodies (2016, methodology): “In [9] this result was unified with the sufficiency part of the subspace concentration condition in the even discrete case by introducing the notation of essential subspaces.”
• Existence of solutions to the even dual Minkowski problem (2018, background): “See also Jian, Lu & Zhu [35], Lu & Wang [36], Zhu [71], etc., on this problem.”
• The dual Minkowski problem for negative indices (2017, methodology): “Böröczky, Hegedűs, & Zhu [5] later found a sufficient condition for the discrete case, which includes [9] (in the discrete case) and [49] as special cases.”
• Necessary subspace concentration conditions for the even dual Minkowski problem (2017, unknown): “In [15], Chen, Li and Zhu proved that also in the non-symmetric logarithmic Minkowski problem the subspace concentration condition is sufficient.”