724 research outputs found
Trisections of a 3-rotationally symmetric planar convex body minimizing the maximum relative diameter
In this work we study the fencing problem consisting of finnding a trisection
of a 3-rotationally symmetric planar convex body which minimizes the maximum
relative diameter. We prove that an optimal solution is given by the so-called
standard trisection. We also determine the optimal set giving the minimum value
for this functional and study the corresponding universal lower bound.Comment: Preliminary version, 20 pages, 15 figure
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
Cubic Partial Cubes from Simplicial Arrangements
We show how to construct a cubic partial cube from any simplicial arrangement
of lines or pseudolines in the projective plane. As a consequence, we find nine
new infinite families of cubic partial cubes as well as many sporadic examples.Comment: 11 pages, 10 figure
Statistical hyperbolicity in groups
In this paper, we introduce a geometric statistic called the "sprawl" of a
group with respect to a generating set, based on the average distance in the
word metric between pairs of words of equal length. The sprawl quantifies a
certain obstruction to hyperbolicity. Group presentations with maximum sprawl
(i.e., without this obstruction) are called statistically hyperbolic. We first
relate sprawl to curvature and show that nonelementary hyperbolic groups are
statistically hyperbolic, then give some results for products, for
Diestel-Leader graphs and lamplighter groups. In free abelian groups, the word
metrics asymptotically approach norms induced by convex polytopes, causing the
study of sprawl to reduce to a problem in convex geometry. We present an
algorithm that computes sprawl exactly for any generating set, thus quantifying
the failure of various presentations of Z^d to be hyperbolic. This leads to a
conjecture about the extreme values, with a connection to the classic Mahler
conjecture.Comment: 14 pages, 5 figures. This is split off from the paper "The geometry
of spheres in free abelian groups.
On the average number of normals through points of a convex body
In 1944, Santal\'o asked about the average number of normals through a point
of a given convex body. Since then, numerous results appeared in the literature
about this problem. The aim of this paper is to give a concise summary of these
results, with some new, recent developments. We point out connections of this
problem to static equilibria of rigid bodies as well as to geometric partial
differential equations of surface evolution.Comment: 15 page
Decompositions of a polygon into centrally symmetric pieces
In this paper we deal with edge-to-edge, irreducible decompositions of a
centrally symmetric convex -gon into centrally symmetric convex pieces.
We prove an upper bound on the number of these decompositions for any value of
, and characterize them for octagons.Comment: 17 pages, 17 figure
Shortest closed billiard orbits on convex tables
Given a planar compact convex billiard table , we give an algorithm to
find the shortest generalised closed billiard orbits on . (Generalised
billiard orbits are usual billiard orbits if has smooth boundary.) This
algorithm is finite if is a polygon and provides an approximation scheme in
general. As an illustration, we show that the shortest generalised closed
billiard orbit in a regular -gon is 2-bounce for , with
length twice the width of . As an application we obtain an algorithm
computing the Ekeland-Hofer-Zehnder capacity of the four-dimensional domain in the standard symplectic vector space . Our method
is based on the work of Bezdek-Bezdek and on the uniqueness of the Fagnano
triangle in acute triangles. It works, more generally, for planar Minkowski
billiards.Comment: 16 pages, 11 figure
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