26 research outputs found

    From rr-dual sets to uniform contractions

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    Let MdM^d denote the dd-dimensional Euclidean, hyperbolic, or spherical space. The rr-dual set of given set in MdM^d is the intersection of balls of radii rr centered at the points of the given set. In this paper we prove that for any set of given volume in MdM^d the volume of the rr-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 NN congruent balls in Euclidean dd-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 NN sufficiently large) in MdM^d.Comment: 8 page

    Spindle Starshaped Sets

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    In this paper, spindle starshaped sets are introduced and investigated, which apart from normalization form an everywhere dense subfamily within the family of starshaped sets. We focus on proving spindle starshaped analogues of recent theorems of Bobylev, Breen, Toranzos, and Zamfirescu on starshaped sets. Finally, we consider the problem of guarding treasures in an art gallery (in the traditional linear way as well as via spindles).Comment: 16 pages, 2 figure

    Density bounds for outer parallel domains of unit ball packings

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    We give upper bounds for the density of unit ball packings relative to their outer parallel domains and discuss their connection to contact numbers. Also, packings of soft balls are introduced and upper bounds are given for the fraction of space covered by them.Comment: 22 pages, 1 figur
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