The dual Minkowski problem for negative indices

Zhao, Yiming

doi:10.1007/s00526-017-1124-x

Cited by 92 publications

(51 citation statements)

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“…where the first set of equations in (35) hold for (25) follows easily from (23), by first decomposing the integrations over ∂(K ∩ L) and ∂(K ∪ L) into six contributions via (26) and (27), using (30-35), and then recombining these contributions via (28) and (29).…”

Section: General Dual Volumes and Curvature Measuresmentioning

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: General Dual Volumes and Curvature Measuresmentioning

confidence: 99%

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

et al. 2018

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“…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 by Zhao [38], solutions for even µ in [4,6,15,39], and solutions via curvature flows and partial differential equations in [8,24,26].An important extension of the dual Minkowski problem was carried out by Lutwak, Yang, and Zhang [33], who introduced L p dual curvature measures and posed a corresponding L p dual Minkowski problem. In [33], the L 0 addition in [20] is replaced by L p addition, while the qth dual volume remains unchanged.…”

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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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“…Much work has been devoted to these problems. We refer to, e.g., [4,5,6,11,22,39,46,57] for background and progress.…”

Section: Introductionmentioning

confidence: 99%

A Steiner formula in the L Brunn Minkowski theory

Tatarko

Werner

2019

Advances in Mathematics

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We prove an analogue of the classical Steiner formula for the Lp affine surface area of a Minkowski outer parallel body for any real parameters p. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature. They have the intrinsic volumes of the classical Brunn Minkowski theory and the dual quermassintegrals of the dual Brunn Minkowski theory as well as special cases.Properties of these new quantities are investigated, a connection to information theory among them. A Steiner formula for the s-th mixed Lp affine surface area of a Minkowski outer parallel body for any real parameters p and s is also given. *

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The dual Minkowski problem for negative indices

Cited by 92 publications

References 50 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

A Steiner formula in the L Brunn Minkowski theory

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