The dual Minkowski problem for symmetric convex bodies

By K. Boroczky, Erwin Lutwak, Deane Yang, Gaoyong Zhang, Yiming Zhao. Advances in Mathematics, 2017

91 citations recorded by the source corpus.

The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in $\mathbb{R}^n$. A full solution to this is given when $1 < q < n$. The necessary and sufficient condition is an explicit measure concentration condition. A variational approach is used, where the functional is the sum of a dual quermassintegral and an entropy integral. The proof requires two crucial estimates. The first is an estimate of the entropy integral proved using a spherical partition. The second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body.

Subfields: Applied Mathematics, Pharmacology

View Paper: https://www.semanticscholar.org/paper/211e6fce8f8c689fba667b289f8e30ba7888347a

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In Applied Mathematics

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

In Pharmacology

1. Dongmeng Xi
2. Yong Huang
3. Yiming Zhao

Citations in the corpus

• The Minkowski problem in Gaussian probability space (2020, background): “The dual Minkowski problem, while still largely open, has been solved in the o-symmetric case when the associated index q satisfies q ∈ [0, n], see, for example [8, 11, 15, 35, 38, 49, 74, 75].”