Banakh spaces and their geometry

Taras Banakh

Published 2023-05-12Version 1

Following a suggestion of WIll Brian, we define a metric space $(X,d)$ to be $Banakh$ if for every point $c\in X$ and real number $r\in d[X^2]$ the sphere $S(c;r):=\{x\in X:d(x,c)=r\}$ is equal to $\{x,y\}$ for some points $x,y\in X$ with $d(x,y)=2r$. We prove that for any points $x,y\in X$ of a Banakh space with $r:=d(x,y)$, there exists a unique isometric embedding $\ell_{xy}:r\mathbb Z\to X$ such that $\ell_{xy}(0)=x$ and $\ell_{xy}(r)=y$. Using this embedding result, we prove that a metric space $X$ is isometric to a subgroup $G$ of the additive group $\mathbb Q$ of rational numbers if and only if $X$ is a Banakh space such that $d[X^2]=G_+:=\{x\in G:x\ge 0\}$. Also we prove that a metric space $X$ is isometric to the real line if and only if $X$ is a complete Banakh space such that $G_+\subseteq d[X^2]$ for some non-cyclic subgroup $G\subseteq\mathbb Q$. We prove that every Banakh space $X$ with $d[X^2]\subseteq\mathbb Q$ is isometric to some subgroup of $\mathbb Q$. Every Banakh space $X$ satisfies the ``rational'' axiom of segment construction: for every $x,y\in X$ and every $r\in\mathbb Q{\cdot} d(x,y)$ there exists a unique point $z\in X$ such that $d(y,z)=r$ and $d(x,z)=d(x,y)+d(y,z)$. Given a subgroup $G$ of $\mathbb Q$, we show that for every $c\in X$ the $G$-sphere $S(c;G):=\{x\in X:d(c,x)\in G\}$ is isometric to a subgroup of $G$ if and only if the metric space $S(x;G)$ is centrally symmetric in the sense that for any distinct points $a,b\in S(x;G)$ there exists an isometry $f$ of $S(c;G)$ such that $f(a)=a$ and $f(b)\ne b$. Finally, for every nonzero cardinal $\kappa\le\mathfrak c$ in the Hilbert space $\ell_2(\kappa)$ we construct a discrete subgroup $H$ which is a complete Banakh space of cardinality $|H|=\max\{\kappa,\omega\}$, and a dense $\mathbb Q$-linear subspace $L$ in $\ell_2(\kappa)$ such that $L$ is a Banakh space with $d[L^2]=\mathbb R_+$.