A modular version of the Erdős– Szekeres theorem

2001 ◽  
Vol 38 (1-4) ◽  
pp. 245-260 ◽  
Author(s):  
Gy. Károlyi ◽  
J. Pach ◽  
G. Tóth
Keyword(s):  
Convex Hull ◽  
General Position ◽  
Empty Convex ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Mod P

Bialostocki, Dierker, and Voxman proved that for any n = p +2, there is an integer B(n; p) with the following property. Every set of B(n; p) points in general position in the plane has n points in convex position such that the number of points in the interior of their convex hull is 0 mod p. They conjectured that the same is true for all pairs n = 3, p =2. In this note, we show that every su&ciently large point set determining no triangle with more than one point in its interior has n elements that form the vertex set of an empty convex n-gon. As a consequence, we show that the above conjecture is true for all n =5p=6+O(1).

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

Planar point sets with a small number of empty convex polygons

2004 ◽  
Vol 41 (2) ◽  
pp. 243-269 ◽  
Author(s):  
Imre Bárány ◽  
Pável Valtr
Keyword(s):  
Convex Hull ◽  
General Position ◽  
Point Sets ◽  
Convex Polygons ◽  
Planar Point ◽  
Empty Convex ◽  
Finite Set ◽  
Empty Triangles

A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P. We construct a set of n points in general position in the plane with only ˜1.62n2 empty triangles, ˜1.94n2 empty quadrilaterals, ˜1.02n2 empty pentagons, and ˜0.2n2 empty hexagons.

scholarly journals Enumerating Triangulations of Convex Polytopes

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Sergei Bespamyatnikh
Keyword(s):  
Convex Hull ◽  
Simplicial Complex ◽  
Convex Polytopes ◽  
Three Dimensions ◽  
Finite Point ◽  
Convex Position ◽  
International Audience ◽  
Point Set

International audience A triangulation of a finite point set A in $\mathbb{R}^d$ is a geometric simplicial complex which covers the convex hull of $A$ and whose vertices are points of $A$. We study the graph of triangulations whose vertices represent the triangulations and whose edges represent geometric bistellar flips. The main result of this paper is that the graph of triangulations in three dimensions is connected when the points of $A$ are in convex position. We introduce a tree of triangulations and present an algorithm for enumerating triangulations in $O(log log n)$ time per triangulation.

ON FLIPS IN POLYHEDRAL SURFACES

2002 ◽  
Vol 13 (02) ◽  
pp. 303-311 ◽  
Author(s):  
OSWIN AICHHOLZER ◽  
LYUBA S. ALBOUL ◽  
FERRAN HURTADO
Keyword(s):  
Finite Point ◽  
Polyhedral Surfaces ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set

Let V be a finite point set in 3-space, and let [Formula: see text] be the set of triangulated polyhedral surfaces homeomorphic to a sphere and with vertex set V. Let abc and cbd be two adjacent triangles belonging to a surface [Formula: see text]; the flip of the edge bc would replace these two triangles by the triangles abd and adc. The flip operation is only considered when it does not produce a self-intersecting surface. In this paper we show that given two surfaces S1, [Formula: see text], it is possible that there is no sequence of flips transforming S1 into S2, even in the case that V consists of points in convex position.

An Efficient Improved Algorithm of Convex Hull

2014 ◽  
Vol 602-605 ◽  
pp. 3104-3106
Author(s):  
Shao Hua Liu ◽  
Jia Hua Zhang
Keyword(s):  
Convex Hull ◽  
Line Segment ◽  
Main Idea ◽  
Discrete Point ◽  
Directed Line ◽  
Generation Algorithm ◽  
Simple Program ◽  
Point Set ◽  
High Computational Efficiency ◽  
Improved Algorithm

This paper introduced points and directed line segment relation judgment method, the characteristics of generation and Graham method using the original convex hull generation algorithm of convex hull discrete points of the convex hull, an improved algorithm for planar discrete point set is proposed. The main idea is to use quadrilateral to divide planar discrete point set into five blocks, and then by judgment in addition to the four district quadrilateral internally within the point is in a convex edge. The result shows that the method is relatively simple program, high computational efficiency.

scholarly journals Decomposable Convexities in Graphs and Hypergraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Nonempty Subset ◽  
Convex Set ◽  
Convex Geometry ◽  
Convexity Space ◽  
Vertex Set ◽  
Convex Geometries ◽  
Acyclic Hypergraph ◽  
Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

Sphericity Evaluation Based on Minimum Circumscribed Sphere Method

2012 ◽  
Vol 433-440 ◽  
pp. 3146-3151 ◽  
Author(s):  
Fan Wu Meng ◽  
Chun Guang Xu ◽  
Juan Hao ◽  
Ding Guo Xiao
Keyword(s):  
Convex Hull ◽  
Early Stage ◽  
Measured Data ◽  
Efficient Approach ◽  
Critical Data ◽  
Material Condition ◽  
Data Points ◽  
Point Set ◽  
Data Point

The search of sphericity evaluation is a time-consuming work. The minimum circumscribed sphere (MCS) is suitable for the sphere with the maximum material condition. An algorithm of sphericity evaluation based on the MCS is introduced. The MCS of a measured data point set is determined by a small number of critical data points according to geometric criteria. The vertices of the convex hull are the candidates of these critical data points. Two theorems are developed to solve the sphericity evaluation problems. The validated results show that the proposed strategy offers an effective way to identify the critical data points at the early stage of computation and gives an efficient approach to solve the sphericity problems.

An efficient algorithm for finding the minimum norm point in the convex hull of a finite point set in the plane

1994 ◽  
Vol 16 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Naoki Makimoto ◽  
Ikuo Nakagawa ◽  
Akihisa Tamura
Keyword(s):  
Convex Hull ◽  
Efficient Algorithm ◽  
Minimum Norm ◽  
Finite Point ◽  
Point Set
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