The Jung Theorem for spherical and hyperbolic spaces

Dekster, B. V.

doi:10.1007/bf01874495

Cited by 38 publications

(30 citation statements)

References 7 publications

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“…Then the spherical Jung theorem [10] and so, the expression on the right in Lemma 3.5 is well-defined. Let m be a positive integer satisfying…”

Section: An Upper Bound For the Illumination Numbermentioning

confidence: 94%

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

Bezdek

2011

Discrete Comput Geom

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A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a "fat" one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm's theorem and its proof on illuminating convex bodies of constant width to the family of "fat" spindle convex bodies.

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“…Then the spherical Jung theorem [10] and so, the expression on the right in Lemma 3.5 is well-defined. Let m be a positive integer satisfying…”

Section: An Upper Bound For the Illumination Numbermentioning

confidence: 94%

“…(We note that for the purpose of this discussion we use the degree measure for angles following [14].) Thus, using the spherical Jung theorem [10], we obtain that the Gauss image ν(F ) of any face F of B[X] can be covered by a 2-dimensional closed spherical disk of (angular) radius ≤ arcsin…”

Section: Introductionmentioning

confidence: 99%

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

Bezdek

2011

Discrete Comput Geom

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“…First, we give a lower bound for (8). Jung's theorem ( [7]) implies in a straightforward way that crQ ≤ 2d d+1…”

Section: Proof Of (I) In Theoremmentioning

confidence: 99%

From r-dual sets to uniform contractions

Bezdek

2017

Aequat. Math.

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Let M d denote the d-dimensional Euclidean, hyperbolic, or spherical space. The r-dual set of given set in M d is the intersection of balls of radii r centered at the points of the given set. In this paper we prove that for any set of given volume in M d the volume of the r-dual set becomes maximal if the set is a ball. As an application we prove the following. The Kneser-Poulsen Conjecture states that if the centers of a family of N congruent balls in Euclidean d-space is contracted, then the volume of the intersection does not decrease. A uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. We prove the Kneser-Poulsen conjecture for uniform contractions (with N sufficiently large) in M d . * Keywords and phrases: Kneser-Poulsen conjecture, volume of intersections of balls, Blaschke-Santaló inequality, r-dual set, uniform contraction, Euclidean, hyperbolic, and spherical space.

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“…First, estimate (1) was accompanied in [9] by a lower bound of the dimension of X in terms of the ratio D/R. Second, (1) was extended in [10] to the hyperbolic and the spherical n-space. Here, in the Theorem, we extend estimate (1) further, to a class of metric spaces introduced and studied first by A. D. Alexandrov [3,4].…”

Section: Introductionmentioning

confidence: 99%

The Jung Theorem in metric spaces of curvature bounded above

Dekster

1997

Proc. Amer. Math. Soc.

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Abstract. The classical Jung Theorem states in essence that the diameter D of a compact set X in E n satisfies D ≥ R[(2n + 2)/n] 1/2 where R is the circumradius of X. The theorem was extended recently to the hyperbolic and the spherical n-spaces. Here, the estimate above is extended to a class of metric spaces of curvature ≤ K introduced by A. D. Alexandrov. The class includes the Riemannian spaces. The extended estimate is of the form D ≥ f (R, K, n) where n is a positive integer suitably defined for the set X and its circumcenter. It can be that n is not unique or does not exist. In the latter case, no estimate is derived. In case of a Riemannian d-dimensional space, an integer n always exists and satisfies n ≤ d.

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The Jung Theorem for spherical and hyperbolic spaces

Cited by 38 publications

References 7 publications

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

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

From r-dual sets to uniform contractions

The Jung Theorem in metric spaces of curvature bounded above

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