Subspace concentration of dual curvature measures of symmetric convex bodies

Böröczky, Károly J.; Henk, Martin; Pollehn, Hannes

doi:10.4310/jdg/1531188189

Cited by 84 publications

(54 citation statements)

References 40 publications

(44 reference statements)

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“…Here h and ρ are the support function and radial function of M respectively. Now, we show that the functional J(M t ) is non-increasing along the flow (6). Proof.…”

Section: Now Recalling Lemma 1 and Corollary 1 One Can Easily See Thmentioning

confidence: 80%

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A flow method for the dual Orlicz–Minkowski problem

Liu

2020

Trans. Amer. Math. Soc.

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“…Here h and ρ are the support function and radial function of M respectively. Now, we show that the functional J(M t ) is non-increasing along the flow (6). Proof.…”

Section: Now Recalling Lemma 1 and Corollary 1 One Can Easily See Thmentioning

confidence: 80%

“…In this section, we will give a priori estimates about support function and obtain the long-time existence of the flow (6).…”

Section: Long-time Existence Of the Flowmentioning

confidence: 99%

A flow method for the dual Orlicz–Minkowski problem

Liu

2020

Trans. Amer. 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.…”

mentioning

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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“…The dual Minkowski problem miraculously contains problems such as the Aleksandrov problem (q = 0) and the logarithmic Minkowski problem (q = n) as special cases. The problem quickly became the center of attention, see, for example, [6,9,13,19,22,25,26,30,40,49,50].…”

Section: Introductionmentioning

confidence: 99%