The Hadamard variational formula and the Minkowski problem for p-capacity

Colesanti, Andrea; Nyström, Kaj; Salani, Paolo; Xiao, Jie; Yang, Deane; Zhang, Gaoyong

doi:10.1016/j.aim.2015.06.022

Cited by 71 publications

(86 citation statements)

References 87 publications

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“…It has been proved in [34, Propositions 3.1 and 3.3] that the solutions to the polar Orlicz-Minkowski problem for discrete measures must be polytopes, the convex hulls of finite points in R n . It is well-known that all convex bodies can be approximated by polytopes, and hence to study the Minkowski type problems for discrete measures is very important and receives extensive attention, see e.g., [2,3,11,15,21,23,26,29,30,53,65,66,67].…”

Section: The General Dual-polar Orlicz-minkowski Problem For Discretementioning

confidence: 99%

“…The surface area measure S K may be replaced by other measures; for instance, Luo, Ye and Zhu in [34] obtained the p-capacitary Orlicz-Petty bodies where the surface area measure is replaced by the p-capacitary measure (see e.g., [11,28]). As explained in [34], the polar Orlicz-Minkowski problem (i.e., Problems 4.1 and 5.3 with G = t n /n) and the optimization problems (5.13) and (5.14) are quite different in their general forms; however these two problems are also very closely related.…”

Section: The General Orlicz-petty Bodiesmentioning

confidence: 99%

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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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Section: The General Dual-polar Orlicz-minkowski Problem For Discretementioning

confidence: 99%

Section: The General Orlicz-petty Bodiesmentioning

confidence: 99%

On the general dual Orlicz-Minkowski problem

Xing¹,

Ye²

2020

Indiana Univ. Math. J.

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“…Now we provide some basic background for the p-capacity, and all results can be found in e.g. [8,9,10]. Let C ∞ c (R n ) be the set of all infinitely differentiable functions on R n with compact supports.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

“…where a + = max{a, 0} for all a ∈ R. The p-capacity of K ∈ K 0 can be calculated by the famous Poincaré formula (see e.g. [8]):…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

On the polar Orlicz-Minkowski problems and the p-capacitary Orlicz-Petty bodies

Luo¹

2020

Indiana Univ. Math. J.

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In this paper, we propose and study the polar Orlicz-Minkowski problems: under what conditions on a nonzero finite measure µ and a continuous function ϕ : (0, ∞) → (0, ∞), there exists a convex body K ∈ K 0 such that K is an optimizer of the following optimization problems:The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on ϕ, the existence of a solution is proved for a nonzero finite measure µ on S n−1 which is not concentrated on any hemisphere of S n−1 . Another part of this paper deals with the p-capacitary Orlicz-Petty bodies. In particular, the existence of the p-capacitary Orlicz-Petty bodies is established and the continuity of the p-capacitary Orlicz-Petty bodies is proved.

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“…2-7.3-7.4 in[14] are still valid for the(1,2] ∋ p-equilibrium potential uΩ.Secondly, [22, Lemma 6.16] can be used to produce two constants c > 0 and ǫ ∈ (0, 1) (depending on the Lipschitz constant of Ω) such that (cf [14, . Lemma 7.5] for p ∈ (1, 2) and[22, Theorem 6.5] for p = 2) Thirdly, from[14, Lemma 7.7] it follows that if…”

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confidence: 99%

Prescribing Capacitary Curvature Measures on Planar Convex Domains

Xiao

2019

J Geom Anal

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For p ∈ (1, 2] and a bounded, convex, nonempty, open set Ω ⊂ R 2 let µ p (Ω, ·) be the p-capacitary curvature measure (generated by the closureΩ of Ω) on the unit circle S 1 . This paper shows that such a problem of prescribing µ p on a planar convex domain: "Given a finite, nonnegative, Borel measure µ on S 1 , find a bounded, convex, nonempty, open set Ω ⊂ R 2 such that dµ p (Ω, ·) = dµ(·)" is solvable if and only if µ has centroid at the origin and its support supp(µ) does not comprise any pair of antipodal points. And, the solution is unique up to translation. Moreover, if dµ p (Ω, ·) = ψ(·) dℓ(·) with ψ ∈ C k,α and dℓ being the standard arc-length element on S 1 , then ∂Ω is of C k+2,α .

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The Hadamard variational formula and the Minkowski problem for p-capacity

Cited by 71 publications

References 87 publications

On the general dual Orlicz-Minkowski problem

On the general dual Orlicz-Minkowski problem

On the polar Orlicz-Minkowski problems and the p-capacitary Orlicz-Petty bodies

Prescribing Capacitary Curvature Measures on Planar Convex Domains

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