K-distance set

Let $X$ be a set with metric $d$ , $Y\subseteq X$ , and $L=\{d(x,y):x,y\in Y,x\neq y\}$ . 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 $x\in X\setminus Y$ , there exists $y\in Y$ such that $d(x,y)\notin L$ . 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 $X=\mathbb{R}^{2}$ with $d=$ the box metric: $d(x,y)=\max\{|x_{1}-y_{1}|,|x_{2}-y_{2}|\}$ with $x_{i},y_{i}$ 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 $\Delta_{K}(Y)$ , 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