The 𝐿_{𝑝} Aleksandrov problem for origin-symmetric polytopes

Zhao, Yiming

doi:10.1090/proc/14568

Cited by 13 publications

(6 citation statements)

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“…In this case, the solution to the Gauss Image problem is shown to be unique up to a dilation. The L p analogues of the Aleksandrov problem were considered by Huang, Lutwak, Yang and Zhang in [21], by Mui in [32], and by Zhao in [50]. The L p analogue of the Gauss Image Problem was considered in [46] by C. Wu, D. Wu, and Xiang.…”

Section: Introductionmentioning

confidence: 99%

The Discrete Gauss Image Problem

Semenov¹

2022

Preprint

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The Gauss Image problem is a generalization to the question originally posed by Aleksandrov, who studied the existence of the convex body with prescribed Aleksandrov's integral curvature. A simple discrete case of the Gauss Image Problem can be formulated as follows: given a finite set of directions in R n and the same number of unit vectors, does there exist a convex polytope in R n containing the origin in its interior with vertices at given directions such that each normal cone at the vertex contains exactly one of the given vectors in its interior?We pose a combinatorial problem, called the Assignment Problem, for discrete measures. It is shown that the discrete Gauss Image Problem and the Assignment Problem are equivalent. We establish a geometric condition for measures which solves the Assignment Problem and, hence, the Gauss Image Problem. We also show that generically, the Assignment Problem has a solution. The proper reformulation for the uniqueness questions is also addressed and analyzed. The work establishes interesting connections of the Discrete Gauss Image problem to Hall's marriage theorem and transportation polytopes.

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Section: Introductionmentioning

confidence: 99%

The Discrete Gauss Image Problem

Semenov¹

2022

Preprint

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“…Different proofs of the Aleksandrov problem were given by Oliker [34] and Bertand [8]. The L p analogues of the Aleksandrov problem were considered by Huang, Lutwak, Yang and Zhang in [18], by Mui in [30], and by Zhao in [42].…”

Section: Introductionmentioning

confidence: 99%

The Gauss Image Problem with Weak Aleksandrov Condition

Semenov¹

2022

Preprint

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We introduce a relaxation to the Aleksandrov relation assumption for the Gauss Image Problem. This new assumption turns out to be a necessary assumption for two measures to be related by a convex body. We provide several properties of the new condition. A solution to the Gauss Image Problem is obtained for the case of when one of the measures is assumed to be discrete and another measure is assumed to be absolutely continuous, under the new relaxed assumption.

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“…When G = t q /n and ϕ(t) = t p for 0 = p ∈ R, the general dual Orlicz-Minkowski problem reduces to the L p dual Minkowski problem [50]; contributions to this problem can be seen in, e.g., [3,10,11,14,33,38,41,55]. By letting G(u, t) = log t for all (u, t) ∈ S n−1 × (0, ∞), V G (K) for K ∈ K n (o) reduces to the dual entropy of K; in this case one can get the (L p and Orlicz) Aleksandrov problems [1,20,31] (see also [42,68]). Lastly, the general dual Orlicz-Minkowski problem also extends the dual Orlicz-Minkowski problems [65,69] and the Minkowski problem for Gaussian measures [32].…”

Section: Introductionmentioning

confidence: 99%

On the Musielak-Orlicz-Gauss image problem

Huang,

Xing,

et al. 2021

Preprint

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In the present paper we initiate the study of the Musielak-Orlicz-Brunn-Minkowski theory for convex bodies. In particular, we develop the Musielak-Orlicz-Gauss image problem aiming to characterize the Musielak-Orlicz-Gauss image measure of convex bodies. For a convex body K, its Musielak-Orlicz-Gauss image measure, denoted by C Θ (K, •), involves a triple Θ = (G, Ψ, λ) where G and Ψ are two Musielak-Orlicz functions defined on S n−1 × (0, ∞) and λ is a nonzero finite Lebesgue measure on the unit sphere S n−1 . Such a measure can be produced by a variational formula of V G,λ (K) (the general dual volume of K with respect to λ) under the perturbations of K by the Musielak-Orlicz addition defined via the function Ψ. The Musielak-Orlicz-Gauss image problem contains many intensively studied Minkowski type problems and the recent Gauss image problem as its special cases. Under the condition that G is decreasing on its second variable, the existence of solutions to this problem is established.

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The 𝐿_{𝑝} Aleksandrov problem for origin-symmetric polytopes

Abstract: The L p Aleksandrov integral curvature and its corresponding characterization problemthe L p Aleksandrov problem-were recently introduced by Huang, Lutwak, Yang, and Zhang. The current work presents a solution to the L p Aleksandrov problem for origin-symmetric polytopes when −1 < p < 0.

Cited by 13 publications

References 53 publications

The Discrete Gauss Image Problem

The Discrete Gauss Image Problem

The Gauss Image Problem with Weak Aleksandrov Condition

On the Musielak-Orlicz-Gauss image problem

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