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 Disjoint empty convex polygons in planar point sets

2001 ◽  
Vol 56 (2) ◽  
pp. 62-70
Author(s):  
Attila Guly�s ◽  
L�zl� Szab�
Keyword(s):  
Point Sets ◽  
Convex Polygons ◽  
Planar Point ◽  
Empty Convex

scholarly journals Disjoint empty convex pentagons in planar point sets

2012 ◽  
Vol 66 (1) ◽  
pp. 73-86 ◽  
Author(s):  
Bhaswar B. Bhattacharya ◽  
Sandip Das
Keyword(s):  
Point Sets ◽  
Planar Point ◽  
Empty Convex

scholarly journals EMPTY CONVEX 5-GONS IN PLANAR POINT SETS

2009 ◽  
Vol 31 (3) ◽  
pp. 315-321
Author(s):  
Seong-Yoon Ann ◽  
En-Sil Kang
Keyword(s):  
Point Sets ◽  
Planar Point ◽  
Empty Convex

Counting convex polygons in planar point sets

1995 ◽  
Vol 56 (1) ◽  
pp. 45-49 ◽  
Author(s):  
Joseph S.B. Mitchell ◽  
Günter Rote ◽  
Gopalakrishnan Sundaram ◽  
Gerhard Woeginger
Keyword(s):  
Point Sets ◽  
Convex Polygons ◽  
Planar Point

scholarly journals Counting Non-Convex 5-Holes in a Planar Point Set

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 78
Author(s):  
Young-Hun Sung ◽  
Sang Won Bae
Keyword(s):  
General Position ◽  
Brute Force ◽  
Expected Number ◽  
Planar Point ◽  
Point Set ◽  
Empty Triangles ◽  
Bounded Body

Let S be a set of n points in the general position, that is, no three points in S are collinear. A simple k-gon with all corners in S such that its interior avoids any point of S is called a k-hole. In this paper, we present the first algorithm that counts the number of non-convex 5-holes in S. To our best knowledge, prior to this work there was no known algorithm in the literature except a trivial brute force algorithm. Our algorithm runs in time O(T+Q), where T denotes the number of 3-holes, or empty triangles, in S and Q that denotes the number of non-convex 4-holes in S. Note that T+Q ranges from Ω(n2) to O(n3), while its expected number is Θ(n2logn) when the points in S are chosen uniformly and independently at random from a convex and bounded body in the plane.

scholarly journals On empty convex polygons in a planar point set

2006 ◽  
Vol 113 (3) ◽  
pp. 385-419 ◽  
Author(s):  
Rom Pinchasi ◽  
Radoš Radoičić ◽  
Micha Sharir
Keyword(s):  
Convex Polygons ◽  
Planar Point ◽  
Empty Convex ◽  
Point Set

Partitioning a Planar Point Set into Empty Convex Polygons

2003 ◽  
pp. 129-134 ◽  
Author(s):  
Ren Ding ◽  
Kiyoshi Hosono ◽  
Masatsugu Urabe ◽  
Changqing Xu
Keyword(s):  
Convex Polygons ◽  
Planar Point ◽  
Empty Convex ◽  
Point Set
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