On Polygons Enclosing Point Sets II (original) (raw)
Related papers
On polygons enclosing point sets
2001
Let R and B be point sets such that R ∪ B is in general position. We say that B is enclosed by R if there is a simple polygon P with vertex set R such that all the elements in B belong to the interior of P. In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv (R)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56|Conv (B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p 1 ,. .. , p n and S a set of m points contained in the interior of P , m ≤ n − 1. Then there is a convex decomposition {P 1 ,. .. , P n } of P such that all points from S lie on the boundaries of P 1 ,. .. , P n , and each P i contains a whole edge of P on its boundary.
On Polygons Excluding Point Sets
Graphs and Combinatorics, 2012
By a polygonization of a finite point set S in the plane we understand a simple polygon having S as the set of its vertices. Let B and R be sets of blue and red points, respectively, in the plane such that B ∪ R is in general position, and the convex hull of B contains k interior blue points and l interior red points. Hurtado et al. found sufficient conditions for the existence of a blue polygonization that encloses all red points. We consider the dual question of the existence of a blue polygonization that excludes all red points R. We show that there is a minimal number K = K(l), which is a polynomial in l, such that one can always find a blue polygonization excluding all red points, whenever k ≥ K. Some other related problems are also considered.
The maximum size of a convex polygon in a restricted set of points in the plane
Discrete and Computational …, 1989
Assume we have k points in general position in the plane such that the ratio between the maximum distance of any pair of points to the minimum distance of any pair of points is at most crx/k, for some positive constant a. We show that there exist at least flk 1/4 of these points which are the vertices of a convex polygon, for some positive constant /3 =/3(a). On the other hand, we show that for every fixed e>0, if k>k(e), then there is a set of k points in the plane for which the above ratio is at most 4~, which does not contain a convex polygon of more than k 1/3+~ vertices.
Covering convex sets with non-overlapping polygons
Discrete Mathematics, 1990
We prove that given n 2 3 convex, compact, and pairwise disjoint sets in the plane, they may be covered with n non-overlapping convex polygons with a total of not more than 6n -9 sides, and with not more than 3n -6 distinct slopes. Furthermore, we construct sets that require 6n -9 sides and 3n -6 slopes for n 2 3. The upper bound on the number of slopes implies a new bound on a recently studied transversal problem.
Computational Geometry, 2012
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On Weighted Sums of Numbers of Convex Polygons in Point Sets
Discrete & Computational Geometry
Let S be a set of n points in general position in the plane, and let X_{k,\ell }(S)Xk,ℓ(S)bethenumberofconvexk−gonswithverticesinSthathaveexactlyX k , ℓ ( S ) be the number of convex k-gons with vertices in S that have exactlyXk,ℓ(S)bethenumberofconvexk−gonswithverticesinSthathaveexactly\ell ℓpointsofSintheirinterior.Weproveseveralequalitiesforthenumbersℓ points of S in their interior. We prove several equalities for the numbersℓpointsofSintheirinterior.WeproveseveralequalitiesforthenumbersX_{k,\ell }(S)$$ X k , ℓ ( S ) . This problem is related to the Erdős–Szekeres theorem. Some of the obtained equations also extend known equations for the numbers of empty convex polygons to polygons with interior points. Analogous results for higher dimension are shown as well.
On the Erdos-Szekeres n-interior point problem
2014
The n-interior point variant of the Erdos-Szekeres problem is to show the following: For any n, n ≥ 1, every point set in the plane with sufficient number of interior points contains a convex polygon containing exactly n-interior points. This has been proved only for n ≤ 3. In this paper, we prove it for pointsets having atmost logarithmic number of convex layers. We also show that any pointset containing atleast n interior points, there exists a 2-convex polygon that contains exactly n-interior points.
On the perimeters of simple polygons contained in a plane convex body
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2013
A simple n-gon is a polygon with n edges such that each vertex belongs to exactly two edges and every other point belongs to at most one edge. Brass, Moser and Pach [2] asked the following question: For n ≥ 5 odd, what is the maximum perimeter of a simple n-gon contained in a Euclidean unit disk? In 2009, Audet, Hansen and Messine [1] answered this question, and showed that the supremum is the perimeter of an isosceles triangle inscribed in the disk, with an edge of multiplicity n − 2. In [3], Lángi generalized their result for polygons contained in a hyperbolic disk. In this note we find the supremum of the perimeters of simple n-gons contained in an arbitrary plane convex body in the Euclidean or in the hyperbolic plane.
On relatively equilateral polygons inscribed in a convex body
Publicationes Mathematicae Debrecen
Let C ⊂ E 2 be a convex body. The C-length of a segment is the ratio of its length to the half of the length of a longest parallel chord of C. By a relatively equilateral polygon inscribed in C we mean an inscribed convex polygon all of whose sides are of equal C-length. We prove that for every boundary point x of C and every integer k ≥ 3 there exists a relatively equilateral k-gon with vertex x inscribed in C. We discuss the C-length of sides of relatively equilateral k-gons inscribed in C and we reformulate this question in terms of packing C by k homothetical copies which touch the boundary of C. Let C be a convex body in Euclidean n-space E n. If pq is a longest chord of C in a direction l, we say that points p and q are opposite and we call pq a diametral chord of C in direction l. By the C-distance dist C (a, b) of a and b we mean the ratio of the Euclidean distance |ab| of a and b to the half of the Euclidean distance of end-points of a diametral chord of C parallel to ab (comp. [7]). We use here the term relative distance if there is no doubt about C. By the C-length of the segment ab we mean dist C (a, b). If C ⊂ E 2 , we define a C-equilateral k-gon as a convex k-gon all of whose sides have equal C-lengths. We also use the name relatively equilateral k-gon when C is fixed. Section 1 is of an auxiliary nature. It presents properties of the Cdistance, and especially properties of the C-distance of boundary points