26 research outputs found
From -dual sets to uniform contractions
Let denote the -dimensional Euclidean, hyperbolic, or spherical
space. The -dual set of given set in is the intersection of balls of
radii centered at the points of the given set. In this paper we prove that
for any set of given volume in the volume of the -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
congruent balls in Euclidean -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 sufficiently large) in .Comment: 8 page
Spindle Starshaped Sets
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
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