Illuminating Spindle Convex Bodies and Minimizing the Volume of Spherical Sets of Constant Width

Bezdek, Károly

doi:10.1007/s00454-011-9369-1

Cited by 17 publications

(17 citation statements)

References 22 publications

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“…S is a self-dual sector when S * = S. We refer to [10] for the definition of sectors of constant width. In Section 4, an explicit definition is given for sectors in S 2 .…”

Section: A Variational Problem On Sectorsmentioning

confidence: 99%

On variational problems related to steepest descent curves and self-dual convex sets on the sphere

Longinetti

Manselli

Venturi

2014

Applicable Analysis

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Let C be the family of compact convex subsets S of the hemisphere in R n with the property that S contains its dual S * ; let u ∈ S * , and let (S, u) = 2 ω n S θ, u dσ (θ). The problem to study inf (S, u), S ∈ C, u ∈ S * is considered. It is proved that there exists a minimizing couple (S, u) such that S is self-dual and u is on its boundary. More can be said for n = 3: the minimum set is a Reuleaux triangle on the sphere. The previous problem is related to the one to find the maximal length of steepest descent curves for quasi-convex functions, satisfying suitable constraints. For n = 2, let us refer to [Manselli P, Pucci C. Maximum length of steepest descent curves for quasi-convex functions. Geom. Dedicata. 1991]. Here, quite different results are obtained for n ≥ 3.

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“…S is a self-dual sector when S * = S. We refer to [10] for the definition of sectors of constant width. In Section 4, an explicit definition is given for sectors in S 2 .…”

Section: A Variational Problem On Sectorsmentioning

confidence: 99%

On variational problems related to steepest descent curves and self-dual convex sets on the sphere

Longinetti

Manselli

Venturi

2014

Applicable Analysis

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“…If an exposed b-face is a singleton {x 0 }, then x 0 is called a b-exposed point of K, and b-exp(K) denotes the set of all b-exposed points. We note that several such concepts, referring to the analogous notions for ball polytopes, their boundary structure, separation properties with respect to spheres etc., can be found in the papers [7,25], but are defined there only for the subcase of the Euclidean norm. Finally, a set S is called b-bounded if rad(S) < 1.…”

Section: B(x 1)mentioning

confidence: 99%

Ball convex bodies in Minkowski spaces

2017

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Abstract. The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset S of a normed space is called ball convex if it coincides with its ball hull, which is obtained as intersection of all balls (of fixed radius) containing S. Ball convex sets are closely related to notions like ball polytopes, complete sets, bodies of constant width, and spindle convexity. We will study geometric properties of ball convex bodies in normed spaces, for example deriving separation theorems, characterizations of strictly convex norms, and an application to complete sets. Our main results refer to minimal representations of ball convex bodies in terms of their ball exposed faces, to representations of ball hulls of sets via unions of ball hulls of finite subsets, and to ball convexity of increasing unions of ball convex bodies.

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“…From the characterization of the equality cases in (11), this only converges to equality when we consider sequences of sets (K • n ) n and (−L • n ) n converging to simplices with 0 in one of the vertices and the same outer normal vectors at the facets that pass through the origin.…”

Section: Colesanti's Inequality For Two Functionsmentioning

confidence: 99%

Rogers–Shephard inequality for log-concave functions

Alonso–Gutiérrez

Jimenez

Merino

et al. 2016

Journal of Functional Analysis

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In this paper we prove different functional inequalities extending the classical Rogers-Shephard inequalities for convex bodies. The original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body, such as, its difference body. We characterize the equality cases in all these inequalities. Our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log-concave functions and the study of some geometrical properties of these new sets.

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Illuminating Spindle Convex Bodies and Minimizing the Volume of Spherical Sets of Constant Width

Cited by 17 publications

References 22 publications

On variational problems related to steepest descent curves and self-dual convex sets on the sphere

On variational problems related to steepest descent curves and self-dual convex sets on the sphere

Ball convex bodies in Minkowski spaces

Rogers–Shephard inequality for log-concave functions

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