The Dual Orlicz–Minkowski Problem

Zhu, Baocheng; Xing, Sudan; Ye, Deping

doi:10.1007/s12220-018-0002-x

Cited by 39 publications

(31 citation statements)

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“…Here we introduce very general dual Orlicz curvature measures which specialize to both the L p dual curvature measures in [21] and the dual Orlicz curvature measures defined in [24,27]. We state the corresponding Minkowski problem and present a partial solution, though one general enough to include those from [24,27] as well as solving the case p > 0 and q < 0 of the Minkowski problem posed in [21,Problem 8.1]. (After we proved our result, we learned that Böröczky and Fodor [2] have solved the case p > 1 and q > 0.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2018

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The general volume of a star body, a notion that includes the usual volume, the qth dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the (p, q)-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski problem posed by Lutwak, Yang, and Zhang for the (p, q)-dual curvature measures when p > 0 and q < 0. A dual Orlicz-Brunn-Minkowski inequality for general volumes is obtained, as well as dual Orlicz-Minkowski-type inequalities and uniqueness results for star bodies. Finally, a very general Minkowski-type inequality, involving two Orlicz functions, two convex bodies, and a star body, is proved, that includes as special cases several others in the literature, in particular one due to Lutwak, Yang, and Zhang for the (p, q)-mixed volume.

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Section: Introductionmentioning

confidence: 99%

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

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“…Böröczky and Fodor [1] provide a beautiful solution to the L p dual Minkowski problem for general µ when p > 1 and q > 0.The first Orlicz version of the Minkowski problem appeared in [14], at the inception of the Orlicz-Brunn-Minkowski theory in 2010. Later, the dual Minkowski problem was extended to the Orlicz setting and partially solved in [37,41]; here, the qth dual volume in [20] is replaced by certain dual Orlicz quermassintegrals, while L 0 addition is retained. A common generalization of the problems in [33,37,41] was proposed in [12], in which the qth dual volume is replaced by a very general notion of dual volume denoted by V G (K) and, simultaneously, L 0 addition is replaced by an extension of L p addition called Orlicz addition.…”

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confidence: 99%

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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“…Naturally, the corresponding Minkowski problem in the Orlicz Brunn-Minkowski theory is called the Orlicz Minkowski problem which was first investigated by Haberl et al [50] for even measures. Today, great progress has been made on it (see, e.g., [51][52][53][54][55][56][57][58][59][60]). The present paper is aimed at dealing with the Orlicz capacitary Minkowski problem.…”

Section: Introductionmentioning

confidence: 99%

On the Discrete Orlicz Electrostatic $q$ -Capacitary Minkowski Problem

Feng

Zhou

2020

Journal of Function Spaces

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We establish the existence of solutions to the Orlicz electrostatic q -capacitary Minkowski problem for polytopes. This contains a new result of the discrete L p electrostatic q -capacitary Minkowski problem for p < 0 and 1 < q < n .

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The Dual Orlicz–Minkowski Problem

Cited by 39 publications

References 65 publications

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

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

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

On the Discrete Orlicz Electrostatic $q$ -Capacitary Minkowski Problem

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The Dual Orlicz–Minkowski Problem

Cited by 39 publications

References 65 publications

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

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

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

On the Discrete Orlicz Electrostaticq-Capacitary Minkowski Problem

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On the Discrete Orlicz Electrostatic $q$ -Capacitary Minkowski Problem