Asymmetry of Convex Bodies of Constant Width

Jin, Hailin; Guo, Qi

doi:10.1007/s00454-011-9370-8

Cited by 25 publications

(8 citation statements)

References 11 publications

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“…According to [6,Example 4], a set can satisfy (5) and contain segments on the boundary, also when X is the Euclidean plane. We shall prove that the same cannot happen if A satisfies (CR), so extending [13,Lemma 2].…”

Section: Another Conditionmentioning

confidence: 84%

Completions and balls in Banach spaces

Papini¹

2015

Ann. Funct. Anal.

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In this paper we study properties of complete sets and of completions of sets in Banach spaces, with relation to balls: in particular, we study conditions implying that a set is a ball, or it has a completion which is a ball. We also discuss some related simple properties considered in a few recent papers, a new similar property, and relations among classes of sets defined by them. Results are illustrated by several examples.

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Section: Another Conditionmentioning

confidence: 84%

Completions and balls in Banach spaces

Papini¹

2015

Ann. Funct. Anal.

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“…It was known that, for most kinds of measures of asymmetry, the Reuleaux triangle is the most asymmetric one among all domains of constant width. Recently, we proved that this is true also for the well-known Minkowski measure (see [10]). In this paper, we continue to investigate the asymmetry of Reuleaux polygons and prove that, for the Minkowski measure of asymmetry, the regular Reuleaux…”

Section: Introductionmentioning

confidence: 86%

“…In 2-dimensional space R 2 , the convex domains of constant width, in particular the Reuleaux polygons, got much attention (see [1,3,4,7,10,[12][13][14]). It was known that, for most kinds of measures of asymmetry, the Reuleaux triangle is the most asymmetric one among all domains of constant width.…”

Section: Introductionmentioning

confidence: 99%

On a measure of asymmetry for Reuleaux polygons

Guo

Jin

2011

J. Geom.

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In a previous paper, we showed that for regular Reuleaux polygons Rn the equality as∞(Rn) = 1/(2 cos π 2n − 1) holds, where as∞(·) denotes the Minkowski measure of asymmetry for convex bodies, and as∞(K) ≤ 1 2 ( √ 3 + 1) for all convex domains K of constant width, with equality holds iff K is a Reuleaux triangle. In this paper, we investigate the Minkowski measures of asymmetry among all Reuleaux polygons of order n and show that regular Reuleaux polygons of order n (n ≥ 3 and odd) have the minimal Minkowski measure of asymmetry.Mathematics Subject Classification (2010). 52A38, 52A10.

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“…In these two constructions, the regular simplex seems to play a crucial role in the optimality (see also [8] for a more rigorous justification of this intuition). It is therefore natural to search for the body with constant width that is the closest to a regular simplex.…”

Section: Application III : Closest Convex Set With Constant Widthmentioning

confidence: 99%

Handling Convexity-Like Constraints in Variational Problems

Mérigot¹,

Oudet²

2014

SIAM J. Numer. Anal.

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Abstract. We provide a general framework to construct finite dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give precise estimates of the distance between the approximation space and the admissible set. This framework applies to the approximation of convex functions by piecewise linear functions on a mesh of the domain and by other finite-dimensional spaces such as tensor-product splines. We show how these discretizations are well suited for the numerical solution of problems of calculus of variations under convexity constraints. Our implementation relies on proximal algorithms, and can be easily parallelized, thus making it applicable to large scale problems in dimension two and three. We illustrate the versatility and the efficiency of our approach on the numerical solution of three problems in calculus of variation : 3D denoising, the principal agent problem, and optimization within the class of convex bodies.

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Asymmetry of Convex Bodies of Constant Width

Cited by 25 publications

References 11 publications

Completions and balls in Banach spaces

Completions and balls in Banach spaces

On a measure of asymmetry for Reuleaux polygons

Handling Convexity-Like Constraints in Variational Problems

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