The dual Minkowski problem for symmetric convex bodies

Böröczky, Károly J.; Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong; Zhao, Yiming

doi:10.1016/j.aim.2019.106805

Cited by 44 publications

(26 citation statements)

References 57 publications

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“…In [9] and [29], the existence of weak solutions was proved when the inhomogeneous term is a non-negative measure not concentrated in any sub-spaces. For other related results, we refer the readers to [8,10,37] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

Sheng

Wang

2019

J. Eur. Math. Soc.

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In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space R n+1 with speed f r α K, where K is the Gauss curvature, r is the distance from the hypersurface to the origin, and f is a positive and smooth function. If α ≥ n + 1, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if f ≡ 1. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang [29], for the case q < 0. If α < n+1, corresponding to the case q > 0, we also establish the same results for even function f and originsymmetric initial condition, but for non-symmetric f , counterexample is given for the above smooth convergence.2010 Mathematics Subject Classification. 35K96, 53C44.

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Section: Introductionmentioning

confidence: 99%

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

Sheng

Wang

2019

J. Eur. Math. Soc.

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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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“…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%

“…Problem 5. 6. Under what conditions on a nonzero finite Borel measure µ defined on S n−1 , continuous functions ϕ : (0, ∞) → (0, ∞) and G : (0, ∞) × S n−1 → (0, ∞) can we find a convex body K ∈ K n (o) solving the following optimization problems:…”

Section: The Polar Orlicz-minkowski Problem Associated With the Genermentioning

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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The dual Minkowski problem for symmetric convex bodies

Cited by 44 publications

References 57 publications

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

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

On the general dual Orlicz-Minkowski problem

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