25,421 research outputs found
On k-Convex Polygons
We introduce a notion of -convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{-convex} objects considered.Comment: 23 pages, 19 figure
Approximating the Maximum Overlap of Polygons under Translation
Let and be two simple polygons in the plane of total complexity ,
each of which can be decomposed into at most convex parts. We present an
-approximation algorithm, for finding the translation of ,
which maximizes its area of overlap with . Our algorithm runs in
time, where is a constant that depends only on and .
This suggest that for polygons that are "close" to being convex, the problem
can be solved (approximately), in near linear time
Convex lattice polygons of fixed area with perimeter dependent weights
We study fully convex polygons with a given area, and variable perimeter
length on square and hexagonal lattices. We attach a weight t^m to a convex
polygon of perimeter m and show that the sum of weights of all polygons with a
fixed area s varies as s^{-theta_{conv}} exp[K s^(1/2)] for large s and t less
than a critical threshold t_c, where K is a t-dependent constant, and
theta_{conv} is a critical exponent which does not change with t. We find
theta_{conv} is 1/4 for the square lattice, but -1/4 for the hexagonal lattice.
The reason for this unexpected non-universality of theta_{conv} is traced to
existence of sharp corners in the asymptotic shape of these polygons.Comment: 8 pages, 5 figures, revtex
The cross covariogram of a pair of polygons determines both polygons, with a few exceptions
The cross covariogram g_{K,L} of two convex sets K and L in R^n is the
function which associates to each x in R^n the volume of the intersection of K
and L+x.
Very recently Averkov and Bianchi [AB] have confirmed Matheron's conjecture
on the covariogram problem, that asserts that any planar convex body K is
determined by the knowledge of g_{K,K}.
The problem of determining the sets from their covariogram is relevant in
probability, in statistical shape recognition and in the determination of the
atomic structure of a quasicrystal from X-ray diffraction images.
We prove that when K and L are convex polygons (and also when K and L are
planar convex cones) g_{K,L} determines both K and L, up to a described family
of exceptions. These results imply that, when K and L are in these classes, the
information provided by the cross covariogram is so rich as to determine not
only one unknown body, as required by Matheron's conjecture, but two bodies,
with a few classified exceptions.
These results are also used by Bianchi [Bia] to prove that any convex
polytope P in R^3 is determined by g_{P,P}.Comment: 26 pages, 9 figure
Multi-triangulations as complexes of star polygons
Maximal -crossing-free graphs on a planar point set in convex
position, that is, -triangulations, have received attention in recent
literature, with motivation coming from several interpretations of them.
We introduce a new way of looking at -triangulations, namely as complexes
of star polygons. With this tool we give new, direct, proofs of the fundamental
properties of -triangulations, as well as some new results. This
interpretation also opens-up new avenues of research, that we briefly explore
in the last section.Comment: 40 pages, 24 figures; added references, update Section
Central limit theorems for random polygons in an arbitrary convex set
We study the probability distribution of the area and the number of vertices
of random polygons in a convex set . The novel aspect of
our approach is that it yields uniform estimates for all convex sets
without imposing any regularity conditions on the
boundary . Our main result is a central limit theorem for both the
area and the number of vertices, settling a well-known conjecture in the field.
We also obtain asymptotic results relating the growth of the expectation and
variance of these two functionals.Comment: Published in at http://dx.doi.org/10.1214/10-AOP568 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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