scholarly journals A Proof of Erdős-Fishburn's Conjecture for $g(6)=13$

10.37236/2917 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Wei Xianglin
Keyword(s):  
Euclidean Plane ◽  
Interesting Problem ◽  
Planar Point ◽  
Point Set ◽  
Distance Set

A planar point set $X$ in the Euclidean plane is called a $k$-distance set if there are exactly $k$ distances between two distinct points in $X$. An interesting problem is to find the largest possible cardinality of a $k$-distance set. This problem was introduced by Erdős and Fishburn (1996). Maximum planar sets that determine $k$ distances for $k$ less than 5 have been identified. The 6-distance conjecture of Erdős and Fishburn states that 13 is the maximum number of points in the plane that determine exactly 6 different distances. In this paper, we prove the conjecture.

scholarly journals Sparse Distance Sets in the Triangular Lattice

10.37236/3263 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Tanbir Ahmed ◽  
Hunter Snevily
Keyword(s):  
Triangular Lattice ◽  
Euclidean Plane ◽  
Optimal Point ◽  
Planar Point ◽  
Point Configurations ◽  
Point Subset ◽  
Point Set ◽  
Distance Set ◽  
Distance Sets ◽  
Increasing Function

A planar point-set $X$ in Euclidean plane is called a $k$-distance set if there are exactly $k$ different distances among the points in $X$. The function $g(k)$ denotes the maximum number of points in the Euclidean plane that is a $k$-distance set. In 1996, Erdős and Fishburn conjectured that for $k\geq 7$, every $g(k)$-point subset of the plane that determines $k$ different distances is similar to a subset of the triangular lattice. We believe that if $g(k)$ is an increasing function of $k$, then the conjecture is false. We present data that supports our claim and a method of construction that unifies known optimal point configurations for $k\geq 3$.

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.

BASELINE BOUNDED HALF-PLANE VORONOI DIAGRAM

2013 ◽  
Vol 05 (03) ◽  
pp. 1350021 ◽  
Author(s):  
BING SU ◽  
YINFENG XU ◽  
BINHAI ZHU
Keyword(s):  
Voronoi Diagram ◽  
Half Plane ◽  
Line Segment ◽  
Euclidean Plane ◽  
Voronoi Region ◽  
Point Set ◽  
Set Of Points

Given a set of points P = {p1, p2, …, pn} in the Euclidean plane, with each point piassociated with a given direction vi∈ V. P(pi, vi) defines a half-plane and L(pi, vi) denotes the baseline that is perpendicular to viand passing through pi. Define a region dominated by piand vias a Baseline Bounded Half-Plane Voronoi Region, denoted as V or(pi, vi), if a point x ∈ V or(pi, vi), then (1) x ∈ P(pi, vi); (2) the line segment l(x, pi) does not cross any baseline; (3) if there is a point pj, such that x ∈ P(pj, vj), and the line segment l(x, pj) does not cross any baseline then d(x, pi) ≤ d(x, pj), j ≠ i. The Baseline Bounded Half-Plane Voronoi Diagram, denoted as V or(P, V), is the union of all V or(pi, vi). We show that V or(pi, vi) and V or(P, V) can be computed in O(n log n) and O(n2log n) time, respectively. For the heterogeneous point set, the same problem is also considered.

scholarly journals On the Oβ-hull of a planar point set

2018 ◽  
Vol 68 ◽  
pp. 277-291 ◽  
Author(s):  
Carlos Alegría-Galicia ◽  
David Orden ◽  
Carlos Seara ◽  
Jorge Urrutia
Keyword(s):  
Planar Point ◽  
Point Set

PARAMETRIC POLYMATROID OPTIMIZATION AND ITS GEOMETRIC APPLICATIONS

2002 ◽  
Vol 12 (05) ◽  
pp. 429-443 ◽  
Author(s):  
NAOKI KATOH ◽  
HISAO TAMAKI ◽  
TAKESHI TOKUYAMA
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Transportation Problem ◽  
Minimum Weight ◽  
Planar Point ◽  
Line Arrangement ◽  
Optimal Bound ◽  
Straight Line ◽  
Point Set ◽  
Multiple Levels

We give an optimal bound on the number of transitions of the minimum weight base of an integer valued parametric polymatroid. This generalizes and unifies Tamal Dey's O(k1/3 n) upper bound on the number of k-sets (and the complexity of the k-level of a straight-line arrangement), David Eppstein's lower bound on the number of transitions of the minimum weight base of a parametric matroid, and also the Θ(kn) bound on the complexity of the at-most-k level (the union of i-levels for i = 1,2,…,k) of a straight-line arrangement. As applications, we improve Welzl's upper bound on the sum of the complexities of multiple levels, and apply this bound to the number of different equal-sized-bucketings of a planar point set with parallel partition lines. We also consider an application to a special parametric transportation problem.

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