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K-distance set (Definition)

Let $ X$ be a set with metric $ d$, $ Y\subseteq X$, and $ L=\{d(x,y):x,y\in Y,x\ne y\}$. If $ K:=\char93 (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 centered at $ c$ with radius $ r$.

For example, let $ X=\mathbb{R}^2$ with $ d=$ the box metric: $ d(x,y)=\max\{\vert x_1-y_1\vert,\vert x_2-y_2\vert\}$ 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$.



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"K-distance set" is owned by rspuzio. [ full author list (2) | owner history (1) ]
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Cross-references: components, radius, subset, distance, fixed, finite, metric

This is version 3 of K-distance set, born on 2004-04-21, modified 2004-11-04.
Object id is 5788, canonical name is KDistanceSet.
Accessed 1380 times total.

Classification:
AMS MSC: 52C35 (Convex and discrete geometry :: Discrete geometry :: Arrangements of points, flats, hyperplanes)
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