K-distance set

Let X be a set with metric d, YX, and L={d(x,y):x,yY,xy}. If K:=#(L) is finite, Y is said to be a K-distance set.

Y is called a maximal K-distance set if and only if for all xXY, there exists yY such that d(x,y)L. That is, if anything is added to Y, it is no longer a K-distance set.

Y is called a spherical K-distance set if and only if Y is a K-distance set and every element of Y is a fixed distance r from some element c, so Y is a subset of the sphere (http://planetmath.org/SphereMetricSpace) centered at c with radius r.

For example, let X=2 with d= the box metric: d(x,y)=max{|x1-y1|,|x2-y2|} with xi,yi components of x,y, respectively. Let Y={(0,0),(1,0),(2,0),(0,1),(1,1),(2,1),(0,2),(1,2),(2,2)}. Then L={1,2}, so K=2, so Y is a 2-distance set.

Note: please do not confuse this definition of K-distance set with ΔK(Y), the K-distance set of Y.

Title K-distance set
Canonical name KdistanceSet
Date of creation 2013-03-22 14:19:17
Last modified on 2013-03-22 14:19:17
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Definition
Classification msc 52C35