Existence of solutions to the even dual Minkowski problem

By Yiming Zhao. Journal of differential geometry, 2018

126 citations recorded by the source corpus.

Recently, Huang, Lutwak, Yang & Zhang discovered the duals of Federer’s curvature measures within the dual Brunn-Minkowski theory and stated the “Minkowski problem” associated with these new measures. As they showed, this dual Minkowski problem has as special cases the Aleksandrov problem (when the index is 0) and the logarithmic Minkowski problem (when the index is the dimension of the ambient space) — two problems that were never imagined to be connected in any way. Huang, Lutwak, Yang & Zhang established sufficient conditions to guarantee existence of solution to the dual Minkowski problem in the even setting. In this work, existence of solution to the even dual Minkowski problem is established under new sufficiency conditions. It was recently shown by Böröczky, Henk & Pollehn that these new sufficiency conditions are also necessary.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/c9ca448d08b218563edcf163bacd384a20bebbc6

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In Applied Mathematics

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

Citations in the corpus

• Necessary subspace concentration conditions for the even dual Minkowski problem (2017, result): “vex body K ∈ Kn e with Ceq(K,·) = µ. Observe that for q = n the inequality (1.9) becomes essentially (1.8). In case that the parameter q is an integer this result was strengthened by Zhao. Theorem1.4([54]). Let q ∈ {1,...,n−1} and let µ be a non-zero, finite, even Borel measure on Sn−1 satisfying µ(S”
• The dual Minkowski problem for symmetric convex bodies (2017, methodology): “However, for a dual quermassintegral of real degree q > 0, choosing an appropriate barrier is considerably more difficult than it was in [30] and [55].”
• The dual Minkowski problem for negative indices (2017, background): “y recently, for the critical integer cases q = 1,2,··· ,n−1, a better sufficient condition for the existence part of the dual Minkowski problem when the given measure is even was given by the author in [48]. Independently and simultaneously, the same condition was presented by B¨oroczky, Henk & Po”
• 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].”
• Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems (2017, background): “duced by Huang, Lutwak, Yang, and Zhang [29] where they proved the existence of symmetric weak solutions for the case q∈ (0,n+1) under some conditions. Their conditions were recently improved by Zhao [44]. For q<0 the existence and uniqueness of weak solution were obtained in [43]. When q= n+1 it”