Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems

By Yong Huang, Erwin Lutwak, Deane Yang, Gaoyong Zhang. 2016

338 citations recorded by the source corpus.

A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f30b360cb8a83cd04ad73f4c7bccae72fc7832f5

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. Deping Ye
5. Haodi Chen
6. Shibing Chen
7. K. Boroczky
8. Deane Yang

In Geometry and Topology

1. Yiming Zhao
2. Deping Ye
3. Qi-Rui Li
4. Yong Huang
5. Deane Yang
6. K. Boroczky
7. Gaoyong Zhang
8. Erwin Lutwak

Citations in the corpus

• The L dual Minkowski problem for p > 1 and q > 0 (2018, positive): “8 of Huang, Lutwak, Yang, Zhang [17] states the following generalization of Minkowski’s formula (1).”
• On the planar dual Minkowski problem (2018, methodology): “The q-th dual curvature measure in [31] is defined as”
• The Minkowski problem in Gaussian probability space (2020, methodology): “Some of the most prominent Minkowski-type problems include the Lp Minkowski problem (see [18, 39, 58]), the logarithmic Minkowski problem (see [13]), and the dual Minkowski problem (see [38]).”
• Variations of a class of Monge–Ampère-type functionals and their applications (2021, methodology): “The Lp dual Minkowski problem includes the classical Minkowski problem as a special case, and unifies the Lp-Minkowski problem and dual Minkowski problem introduced in [25, 31].”
• The $L_p$-Aleksandrov problem for $L_p$-integral curvature (2018, background): “21) in [24]) that α∗ K(u) ∈ ω if and only if u ∈ αK(ω), for almost all u with respect to spherical Lebesgue measure.”
• The Gauss Image Problem (2020, methodology): “Their connections to dual curvature measures and the dual Minkowski problem were discovered in [27].”
• The Dual Orlicz–Minkowski Problem (2017, methodology): “When φ(t) = t with q ∈ R, one gets the q-th dual curvature measure and the q-th dual quermassintegral of K [20].”
• A flow method for the dual Orlicz–Minkowski problem (2020, background): “They have attracted great attention from many researchers, see for example [11, 20, 22, 27, 29, 30, 32, 33, 34, 36, 43, 45, 51, 53, 55, 56, 58, 60, 63, 66, 69] and references therein.”
• The Lp dual Minkowski problem and related parabolic flows (2021, methodology): “[36] Hug, D.; Lutwak, E.; Yang, D.; Zhang, G.: On the Lp Minkowski problem for polytopes.”
• The Hadwiger Theorem on Convex Functions, I (2020, background): “Interesting new valuations keep arising (see, for example, [8,24,31]) and see [1,2,4,6,7,21,22,29,32,35,36,39] for some recent classification results.”