11,741 research outputs found
On -Gons and -Holes in Point Sets
We consider a variation of the classical Erd\H{o}s-Szekeres problems on the
existence and number of convex -gons and -holes (empty -gons) in a set
of points in the plane. Allowing the -gons to be non-convex, we show
bounds and structural results on maximizing and minimizing their numbers. Most
noteworthy, for any and sufficiently large , we give a quadratic lower
bound for the number of -holes, and show that this number is maximized by
sets in convex position
4-Holes in point sets
We consider a variant of a question of Erdos on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes. In particular, we show that for n >= 9, the maximum number of general 4-holes is ((pi)(4)); the minimum number of general 4-holes is at least 5/2 n(2) - circle minus(n); and the maximum number of non-convex 4-holes is at least 1/2 n(3) - circle minus(n(2) logn) and at most 1/2 n(3) - circle minus(n(2)). 2014 (c) Elsevier B.V. All rights reserved.Postprint (author’s final draft
Compatible 4-Holes in Point Sets
Counting interior-disjoint empty convex polygons in a point set is a typical
Erd\H{o}s-Szekeres-type problem. We study this problem for 4-gons. Let be a
set of points in the plane and in general position. A subset of ,
with four points, is called a -hole in if is in convex position and
its convex hull does not contain any point of in its interior. Two 4-holes
in are compatible if their interiors are disjoint. We show that
contains at least pairwise compatible 4-holes.
This improves the lower bound of which is implied by a
result of Sakai and Urrutia (2007).Comment: 17 page
Engineering Art Galleries
The Art Gallery Problem is one of the most well-known problems in
Computational Geometry, with a rich history in the study of algorithms,
complexity, and variants. Recently there has been a surge in experimental work
on the problem. In this survey, we describe this work, show the chronology of
developments, and compare current algorithms, including two unpublished
versions, in an exhaustive experiment. Furthermore, we show what core
algorithmic ingredients have led to recent successes
On the number of perfect lattices
We show that the number of non-similar perfect -dimensional
lattices satisfies eventually the
inequalities for arbitrary
smallstrictly positive
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