On the L dual Minkowski problem

Huang, Yong; Zhao, Yiming

doi:10.1016/j.aim.2018.05.002

Cited by 95 publications

(61 citation statements)

References 66 publications

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“…In Theorem 6.4, we use the same approximation technique to prove a variant of [12,Theorem 6.4] in the case when G t < 0. When ψ(t) = t p , p > 0, and G(t, u) = t q , q < 0, Theorem 6.4 implies [19,Theorem 3.5]. We end Section 6 with Theorem 6.5, a uniqueness result related to Theorem 6.4 under some additional assumptions on the underlying convex bodies.…”

Section: Introductionmentioning

confidence: 93%

“…Let G : (0, ∞) × S n−1 → (0, ∞) be continuous and such that G t is continuous and negative on (0, ∞) × S n−1 . Let 0 < ε 0 < 1 and suppose that (19) holds for v ∈ S n−1 . Suppose that ψ : (0, ∞) → (0, ∞) is continuous, (24) holds, and that ϕ is finite when defined by (31).…”

Section: The General Dual Orlicz Minkowski Problem For Discrete Measuresmentioning

confidence: 99%

“…If ψ : (0, ∞) → (0, ∞) is continuous, define Let G and G t be continuous on (0, ∞) × S n−1 , where G t < 0 and where G t (t, u) = G t (t, −u) for (t, u) ∈ (0, ∞) × S n−1 . Suppose that there is some 0 < ε 0 < 1 such that (19) holds for v ∈ S n−1 . Let ψ : (0, ∞) → (0, ∞) be continuous and suppose that ϕ is finite when defined by (89).…”

Section: Minkowski Problems For Even Measuresmentioning

confidence: 99%

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General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem II

et al. 2019

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The general dual volume V G (K) and the general dual Orlicz curvature measure C G,ψ (K, ·) were recently introduced for functions G : (0, ∞) × S n−1 → (0, ∞) and convex bodies K in R n containing the origin in their interiors. We extend V G (K) and C G,ψ (K, ·) to more general functions G : [0, ∞) × S n−1 → [0, ∞) and to compact convex sets K containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous dependence on the underlying set, are provided. These are required to study a Minkowski-type problem for the dual Orlicz curvature measure. We mainly focus on the case when G and ψ are both increasing, thus complementing our previous work.The Minkowski problem asks to characterize Borel measures µ on S n−1 for which there is a convex body K in R n containing the origin such that µ equals C G,ψ (K, ·), up to a constant. A major step in the analysis concerns discrete measures µ, for which we prove the existence of convex polytopes containing the origin in their interiors solving the Minkowski problem. For general (not necessarily discrete) measures µ, we use an approximation argument. This approach is also applied to the case where G is decreasing and ψ is increasing, and hence augments our previous work. When the measures µ are even, solutions that are originsymmetric convex bodies are also provided under some mild conditions on G and ψ. Our results generalize several previous works and provide more precise information about the solutions of the Minkowski problem when µ is discrete or even.2 RICHARD J. GARDNER, DANIEL HUG, SUDAN XING, AND DEPING YE problem; see e.g. [23,30,32]. The case p = 0 is of particular significance because the L 0 surface area measure, the so-called cone volume measure, is an affine invariant. The L 0 or logarithmic Minkowski problem is challenging and was only solved recently for even measures by Böröczky, Lutwak, Yang, and Zhang [5]. More recent contributions to the logarithmic Minkowski problem are [3,42] and further references and background on the L p Minkowski problem may be found in [20,36].Recent seminal work of Huang, Lutwak, Yang, and Zhang [20] brought new ingredients, the qth dual curvature measures, to the family of Minkowski problems. These measures are obtained via a first-order variation of the qth dual volume with respect to L 0 addition of convex bodies (see [20, Theorem 4.5]), the case q = n being the L 0 surface area. The authors of [20] posed a corresponding Minkowski problem-the dual Minkowski problem-of finding necessary and/or sufficient conditions for a measure µ on S n−1 to be the qth dual curvature measure of some convex body, and they provided a partial solution when µ is even. Clearly, the logarithmic Minkowski problem is a special case of the dual Minkowski problem. Naturally, the dual Minkowski problem has become important for the dual Brunn-Minkowski theory introduced by Lutwak [28,29]. Since [20], progress includes a complete solution for q < 0...

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Section: Introductionmentioning

confidence: 93%

Section: The General Dual Orlicz Minkowski Problem For Discrete Measuresmentioning

confidence: 99%

Section: Minkowski Problems For Even Measuresmentioning

confidence: 99%

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General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem II

et al. 2019

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“…This is an advantage of the flow method used in our proof, compared to the usual variational method. Theorem B was proved by the continuity method in [35], for which the uniqueness of solution to Eq. (4) is crucial.…”

Section: Introductionmentioning

confidence: 99%

A flow method for the dual Orlicz–Minkowski problem

Liu

2020

Trans. Amer. Math. Soc.

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“…Note that the general dual volume V G (·) was used to derive the general dual Orlicz curvature measures and hence plays central roles in establishing the existence of solutions to the recently proposed general dual Orlicz-Minkowski problem [13,15]. When G(t, u) = 1 n t n , one gets V G (K) = V (K), and when G(t, u) = 1 n t q for q = 0, n, V G (K) becomes the qth dual volume V q (K) which plays fundamental roles in the dual Brunn-Minkowski theory [35,36,37] and the L p dual Minkowski problem (see e.g., [2,4,6,7,24,25,42,61]). When G(t, u) = G(t, e 1 ) for all (t, u) ∈ (0, ∞) × S n−1 , V G (K) becomes the dual Orlicz-quermassintegral in [63]; while if G(t, u) = t 0 φ(ru)r n−1 dr or G(t, u) = ∞ t φ(ru)r n−1 dr for some function φ : R n → (0, ∞), then V G (K) becomes the general dual Orlicz quermassintegral in [55].…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

On the general dual Orlicz-Minkowski problem

Xing¹,

Ye²

2020

Indiana Univ. Math. J.

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This paper gives a systematic study to the general dual-polar Orlicz-Minkowski problem (e.g., Problem 4.1). This problem involves the general dual volume V G (·) recently proposed in [13,15] in order to study the general dual Orlicz-Minkowski problem. As V G (·) extends the volume and the qth dual volume, the general dual-polar Orlicz-Minkowski problem is "polar" to the recently initiated general dual Orlicz-Minkowski problem in [13,15] and "dual" to the newly proposed polar Orlicz-Minkowski problem in [34]. The existence, continuity and uniqueness, if applicable, for the solutions to the general dual-polar Orlicz-Minkowski problem are established. Polytopal solutions and/or counterexamples to the general dual-polar Orlicz-Minkowski problem for discrete measures are also provided. Several variations of the general dual-polar Orlicz-Minkowski problem are discussed as well, in particular the one leading to the general Orlicz-Petty bodies.

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On the L dual Minkowski problem

Cited by 95 publications

References 66 publications

General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem II

General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem II

A flow method for the dual Orlicz–Minkowski problem

On the general dual Orlicz-Minkowski problem

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