Existence of solutions to the even dual Minkowski problem

Zhao, Yiming

doi:10.4310/jdg/1542423629

Cited by 79 publications

(37 citation statements)

References 62 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The dual q-Minkowski problem was recently introduced by Huang, Lutwak, Yang, and Zhang [29] where they proved the existence of symmetric weak solutions for the case q ∈ (0, n + 1) under some conditions. Their conditions were recently improved by Zhao [44]. For q < 0 the existence and uniqueness of weak solution were obtained in [43].…”

Section: Introductionmentioning

confidence: 99%

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

2019

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

2019

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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...

show abstract

“…In this section, we provide a solution to the following dual Orlicz-Minkowski problem: under what conditions on ϕ and a given nonzero finite Borel measure µ on S n−1 , there exist a constant τ > 0 and a convex body K (ideally with the origin in its interior) such that µ = τ C ϕ (K, ·)? When ϕ(t) = t q with 0 = q ≤ n, this problem has been investigated in [20,55,56]. For a nonzero finite Borel measure µ on S n−1 , let…”

Section: A Solution To the Dual Orlicz-minkowski Problemmentioning

confidence: 99%

The Dual Orlicz–Minkowski Problem

Zhu

Xing

2018

J Geom Anal

View full text Add to dashboard Cite

In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the dual Orlicz-Minkowski problem regarding the dual Orlicz curvature measure is provided.2010 Mathematics Subject Classification: 53A15, 52B45, 52A39.

show abstract

Existence of solutions to the even dual Minkowski problem

Cited by 79 publications

References 62 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

The Dual Orlicz–Minkowski Problem

Contact Info

Product

Resources

About