scholarly journals A Note on the Minimum Size of a Point Set Containing Three Nonintersecting Empty Convex Polygons

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 447
Author(s):  
Qing Yang ◽  
Zengtai You ◽  
Xinshang You
Keyword(s):  
Pairwise Disjoint ◽  
Convex Polygon ◽  
Minimum Size ◽  
Convex Polygons ◽  
Planar Point ◽  
Empty Convex ◽  
Point Set

Let P be a planar point set with no three points collinear, k points of P be a k-hole of P if the k points are the vertices of a convex polygon without points of P. This article proves 13 is the smallest integer such that any planar points set containing at least 13 points with no three points collinear, contains a 3-hole, a 4-hole and a 5-hole which are pairwise disjoint.

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

scholarly journals On the minimum size of a point set containing a 5-hole and a disjoint 4-hole

2011 ◽  
Vol 48 (4) ◽  
pp. 445-457 ◽  
Author(s):  
Bhaswar Bhattacharya ◽  
Sandip Das
Keyword(s):  
Upper Bound ◽  
Minimum Size ◽  
Convex Hulls ◽  
Type Result ◽  
Convex Polygons ◽  
Empty Convex ◽  
Ramsey Type ◽  
Point Set

Let H(k; l), k ≦ l denote the smallest integer such that any set of H(k; l) points in the plane, no three on a line, contains an empty convex k-gon and an empty convex l-gon, which are disjoint, that is, their convex hulls do not intersect. Hosono and Urabe [JCDCG, LNCS 3742, 117–122, 2004] proved that 12 ≦ H(4, 5) ≦ 14. Very recently, using a Ramseytype result for disjoint empty convex polygons proved by Aichholzer et al. [Graphs and Combinatorics, Vol. 23, 481–507, 2007], Hosono and Urabe [Kyoto CGGT, LNCS 4535, 90–100, 2008] improve the upper bound to 13. In this paper, with the help of the same Ramsey-type result, we prove that H(4; 5) = 12.

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.

On the minimum cardinality of a planar point set containing two disjoint convex polygons

2013 ◽  
Vol 50 (3) ◽  
pp. 331-354
Author(s):  
Liping Wu ◽  
Wanbing Lu
Keyword(s):  
Positive Integer ◽  
Convex Hulls ◽  
Convex Polygons ◽  
Minimum Cardinality ◽  
Planar Point ◽  
Point Set ◽  
Disjoint Convex

Let N(k, l) be the smallest positive integer such that any set of N(k, l) points in the plane, no three collinear, contains both a convex k-gon and a convex l-gon with disjoint convex hulls. In this paper, we prove that N(3, 4) = 7, N(4, 4) = 9, N(3, 5) = 10 and N(4, 5) = 11.

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
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