Metric characterizations of some subsets of the real line

Abstract

A metric space $(X,\mathsf{d})$ is called a {\em subline} if every 3-element subset $T$ of $X$ can be written as $T=\{x,y,z\}$ for some points $x,y,z$ such that $\mathsf{d}(x,z)=\mathsf{d}(x,y)+\mathsf{d}(y,z)$. By a classical result of Menger, every subline of cardinality $\ne 4$ is isometric to a subspace of the real line. A subline $(X,\mathsf{d})$ is called an {\em $n$-subline} for a natural number $n$ if for every $c\in X$ and positive real number $r\in\mathsf{d}[X^2]$, the sphere ${\mathsf S}(c;r):=\{x\in X\colon \mathsf{d}(x,c)=r\}$ contains at least $n$ points. We prove that every $2$-subline is isometric to some additive subgroup of the real line. Moreover, for every subgroup $G\subseteq{\mathbb R}$, a metric space $(X,\mathsf{d})$ is isometric to $G$ if and only if $X$ is a $2$-subline with $\mathsf{d}[X^2]=G_+:= G\cap[0,\infty)$. A metric space $(X,\mathsf{d})$ is called a {\em ray} if $X$ is a $1$-subline and $X$ contains a point $o\in X$ such that for every $r\in\mathsf{d}[X^2]$ the sphere ${\mathsf S}(o;r)$ is a singleton. We prove that for a subgroup $G\subseteq{\mathbb Q}$, a metric space $(X,\mathsf{d})$ is isometric to the ray $G_+$ if and only if $X$ is a ray with $\mathsf{d}[X^2]=G_+$. A metric space $X$ is isometric to the ray ${\mathbb R}_+$ if and only if $X$ is a complete ray such that ${\mathbb Q}_+\subseteq \mathsf{d}[X^2]$. On the other hand, the real line contains a dense ray $X\subseteq{\mathbb R}$ such that $\mathsf{d}[X^2]={\mathbb R}_+$.