  Following Will Brian, we define a metric space $X$ to be $Banakh$ if all
nonempty spheres of positive radius $r$ in $X$ have cardinality $2$ and
diameter $2r$. Standard examples of Banakh spaces are subgroups of the real
line. In this paper we study the geometry of Banakh spaces, characterize Banakh
spaces which are isometric to subgroups of the real line, and also construct
Banakh spaces $(X,d)$ which do not embed into the real line and have a
prescribed distance set $d[X^2]$.
