1,172 research outputs found
Multiple coverings with closed polygons
A planar set is said to be cover-decomposable if there is a constant
such that every -fold covering of the plane with translates of
can be decomposed into two coverings. It is known that open convex polygons are
cover-decomposable. Here we show that closed, centrally symmetric convex
polygons are also cover-decomposable. We also show that an infinite-fold
covering of the plane with translates of can be decomposed into two
infinite-fold coverings. Both results hold for coverings of any subset of the
plane.Comment: arXiv admin note: text overlap with arXiv:1009.4641 by other author
Unsplittable coverings in the plane
A system of sets forms an {\em -fold covering} of a set if every point
of belongs to at least of its members. A -fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body , there exists a constant such that every
-fold covering of the plane with translates of splits into
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every , an
unsplittable -fold covering of the plane with translates of any open convex
body which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets do not misbehave,
they satisfy the conjecture: every -fold covering of any region of the plane
by translates of such a set splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant such that, for
any positive integer , every -fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} sets
Two-sided bounds for the complexity of cyclic branched coverings of two-bridge links
We consider closed orientable 3-dimensional hyperbolic manifolds which are
cyclic branched coverings of the 3-sphere, with branching set being a
two-bridge knot (or link). We establish two-sided linear bounds depending on
the order of the covering for the Matveev complexity of the covering manifold.
The lower estimate uses the hyperbolic volume and results of Cao-Meyerhoff and
Gueritaud-Futer (who recently improved previous work of Lackenby), while the
upper estimate is based on an explicit triangulation, which also allows us to
give a bound on the Delzant T-invariant of the fundamental group of the
manifold.Comment: Estimates improved using recent results of Gueritaud-Futer and
Kim-Ki
Convex Polygons are Self-Coverable
We introduce a new notion for geometric families called self-coverability and
show that homothets of convex polygons are self-coverable. As a corollary, we
obtain several results about coloring point sets such that any member of the
family with many points contains all colors. This is dual (and in some cases
equivalent) to the much investigated cover-decomposability problem
On the Multiple Packing Densities of Triangles
Given a convex disk and a positive integer , let and
denote the -fold translative packing density and the
-fold lattice packing density of , respectively. Let be a triangle.
In a very recent paper, K. Sriamorn proved that
. In this paper, I will show that
.Comment: arXiv admin note: text overlap with arXiv:1412.539
Indecomposable Coverings with Concave Polygons
We show that for any concave polygon that has no parallel sides and for any k, there is a k-fold covering of some point set by the translates of this polygon that cannot be decomposed into two coverings. Moreover, we give a complete classification of open polygons with this property. We also construct for any polytope (having dimension at least three) and for any k, a k-fold covering of the space by its translates that cannot be decomposed into two covering
Survey on Decomposition of Multiple Coverings
The study of multiple coverings was initiated by Davenport and L. Fejes Tóth more than 50 years ago. In 1980 and 1986, the rst named author published the rst papers about decompos-ability of multiple coverings. It was discovered much later that, besides its theoretical interest, this area has practical applications to sensor networks. Now there is a lot of activity in this eld with several breakthrough results, although, many basic questions are still unsolved. In this survey, we outline the most important results, methods, and questions. 1 Cover-decomposability and the sensor cover problem Let P = { Pi | i ∈ I} be a collection of sets in Rd. We say that P is an m-fold covering if every point of Rd is contained in at least m members of P. The largest such m is called the thickness of the covering. A 1-fold covering is simply called a covering. To formulate the central question of this survey succinctly, we need a denition. Denition 1.1. A planar set P is said to be cover-decomposable if there exists a (minimal) constant m = m(P) such that every m-fold covering of the plane with translates of P can be decomposed into two coverings. Note that the above term is slightly misleading: we decompose (partition) not the set P, but a collection P of its translates. Such a partition is sometimes regarded a coloring of the members of P
A characterization of shortest geodesics on surfaces
Any finite configuration of curves with minimal intersections on a surface is
a configuration of shortest geodesics for some Riemannian metric on the
surface. The metric can be chosen to make the lengths of these geodesics equal
to the number of intersections along them.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-17.abs.htm
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