Nonuniqueness of solutions to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math>-Minkowski problem

Jian, Huaiyu; Lu, Jian; Wang, Xu‐Jia

doi:10.1016/j.aim.2015.05.010

Cited by 86 publications

(29 citation statements)

References 22 publications

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“…Alternate proofs were given by Hug-Lutwak-Yang-Zhang [31]. The case of p < 1 is still largely open (see Böröczky-Lutwak-Yang-Zhang [9], Huang-Liu-Xu [28], Jian-Lu-Wang [32], and Zhu [55,57]). For other recent progress on the L p -Minkowski problem, see Böröczky-Trinh [11] and Chen-Li-Zhu [12,13].…”

Section: Introductionmentioning

confidence: 99%

The dual Minkowski problem for symmetric convex bodies

Böröczky

Lutwak

Yang

et al. 2019

Advances in Mathematics

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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 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.

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Section: Introductionmentioning

confidence: 99%

The dual Minkowski problem for symmetric convex bodies

Böröczky

Lutwak

Yang

et al. 2019

Advances in Mathematics

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“…The solution, when p > 1, was given by Chou & Wang [14]. See also Chen [12], Hug-LYZ [33], Lutwak [42], Lutwak & Oliker [44], LYZ [46], Jian, Lu & Wang [34], and Zhu [73,74]. The solution to the L p Minkowski problem has led to some powerful analytic affine isoperimetric inequalities, see, for example, Haberl & Schuster [30], LYZ [45], Wang [63].…”

Section: Introductionmentioning

confidence: 99%

Existence of solutions to the even dual Minkowski problem

Zhao

2018

J. Differential Geom.

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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.

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“…However, the uniqueness for < 1 is difficult and still open. In [8], Jian et al obtained that, for any − − 1 < < 0, there exists a positive function ∈ ∞ (S ) to guarantee that (8) has two different solutions, which means that we need more conditions to consider the uniqueness.…”

Section: Introductionmentioning

confidence: 99%