The dual Minkowski problem for negative indices

By Yiming Zhao. 2017

130 citations recorded by the source corpus.

Recently, the duals of Federer’s curvature measures, called dual curvature measures, were discovered by Huang et al. (Acta Math 216:325–388, 2016). In the same paper, they posed the dual Minkowski problem, the characterization problem for dual curvature measures, and proved existence results when the index, q, is in (0, n). The dual Minkowski problem includes the Aleksandrov problem ($$q=0$$q=0) and the logarithmic Minkowski problem ($$q=n$$q=n) as special cases. In the current work, a complete solution to the dual Minkowski problem whenever $$q<0$$q<0, including both existence and uniqueness, is presented.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a961d90566394d78039f3b05877bcbafbb288239

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. Shibing Chen
6. Haodi Chen
7. Jian Lu
8. K. Boroczky

In Statistics and Probability

1. Qi-Rui Li
2. Shibing Chen
3. Haodi Chen
4. Yong Huang
5. Gaoyong Zhang
6. Deane Yang
7. Erwin Lutwak
8. K. Boroczky

Citations in the corpus

• On the L dual Minkowski problem (2018, background): “The dual Minkowski problem, while still largely open, has been studied in [8, 12, 30, 33, 68, 69].”
• 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].”
• On the planar dual Minkowski problem (2018, background): “2 is obtained in [51] by a variational argument.”
• Necessary subspace concentration conditions for the even dual Minkowski problem (2017, background): “The case q 0 (including uniqueness) was completely settled by Zhao [53].”
• The Dual Orlicz–Minkowski Problem (2017, background): “In this section, we provide a solution to the following dual Orlicz-Minkowski problem: under what conditions on φ and a given nonzero finite Borel measure μ on Sn−1, there exist a constant τ > 0 and a convex body K (ideally with the origin in its interior) such that μ = τC̃φ(K, ·)? When φ(t) = t wit”
• Variations of a class of Monge–Ampère-type functionals and their applications (2021, methodology): “[4, 7, 16, 17, 19, 21, 30, 32] for the Lp-Minkowski problem, and [5, 15, 24, 25, 29, 38] for the dual Minkowski problem.”
• Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems (2017, background): “For q < 0 the existence and uniqueness of weak solution were obtained in [43].”
• The dual Minkowski problem for symmetric convex bodies (2017, background): “[54] Y. Zhao, The dual Minkowski problem for negative indices, Calc.”
• A flow method for the dual Orlicz–Minkowski problem (2020, background): “(3) is the dual Minkowski problem, which was first introduced by [33] and then studied by [6, 14, 29, 31, 42, 64, 65].”
• The L dual Minkowski problem for p > 1 and q > 0 (2018, unknown): “utwak, Yang, Zhang [16], while the case 0 &lt; p &lt; 1 is due to Chen, Li, Zhu [9]; • if p ≥ 0 and q &lt; 0, then µ = Ce p,q(K,·) for some K ∈ Kn o where the case p = 0 (µ = Ceq(K,·)) is due to Zhao [29] (see also Li, Sheng, Wang [21]), and the case p &gt; 0 is due to Huang, Zhao [19] and Gardner e”