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The two largest distances in finite planar sets

✍ Scribed by Katalin Vesztergombi

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
333 KB
Volume
150
Category
Article
ISSN
0012-365X

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✦ Synopsis

We determine all homogenous linear inequalities satisfied by the numbers of occurrences of the two largest distances among n points in the plane.

πŸ“œ SIMILAR VOLUMES

On large distances in planar sets
On large distances in planar sets
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Let n k denote the number of times the kth largest distance occurs among a set S of n points in the Euclidean plane. We prove that n2 ~< 2~n for arbitrary set S. This upper bound is sharp. We consider the set S of n arbitrary points in R 2. We denote the largest distance between two points in S by

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Let n k denote the number of times the kth largest distance occurs among a set S of n points. We show that if S is the set of vertices of a convex polygone in the euclidean plane, then n1+2n2~3n and n2<~n +n 1. Together with the well-known inequality n~<~n and the trivial inequalities n~>~O and n2>~

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Given positive integers m, k, and s with m > ks, let D m,k,s represent the set {1, 2, . . . , m} -{k, 2k, . . . , sk}. The distance graph G(Z, D m,k,s ) has as vertex set all integers Z and edges connecting i and j whenever |i -j| ∈ D m,k,s . The chromatic number and the fractional chromatic number

The Distance to the Intersection of Two
The Distance to the Intersection of Two Convex Sets Expressed by the Distances to Each of Them
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## Abstract Using the distances of a point __x__ to two convex sets we obtain an upper estimation of the distance of __x__ to the intersection of these two sets. Applications to the intersection of point‐to‐set mappings are given.