Minimal and Maximal Distances in Metric Spaces

TL;DR

This paper characterizes which function pairs can be realized as minimal and maximal distances among points in metric spaces, exploring realizability in Euclidean spaces and the limitations of such configurations.

Contribution

It provides a precise characterization of function pairs realizable as minimal and maximal distances, and analyzes their realizability in Euclidean spaces, revealing exponential growth and specific limitations.

Findings

Characterization of realizable function pairs in metric spaces

Exponential growth of realizable pairs in Euclidean spaces with dimension

Existence of functions realizable as maximal distances in R^2 but not as minimal distances in higher dimensions

Abstract

Given functions $f,g: [n] \rightarrow [n]$ do there exist $n$ points $A_1,A_2\ldots A_n$ in some metric space such that $A_{f(i)},A_{g(i)}$ are the points closest and farthest from point $A_i$? In this paper we characterize precisely which pairs of functions have this property. If the metric space is $\mathbb{R}^k$ we show that the maximal number $m(k)$ so that any pair of functions $f,g: [m(k)]\rightarrow [m(k)]$ realizable in some metric space is also realizable in $\mathbb{R}^k$ grows exponentially in $k$. In the final section of this paper we consider what happens when we look at minimal and maximal distances separately. We show that any function $g$ that can be a maximal distance function can also be a maximal distance function in $\mathbb{R}^2$. We also find an interesting family of functions that can be minimal distance functions but not in $\mathbb{R}^k$.