The betweenness relation distinguishes non-similar pairs of concentric circles

Abstract

Two subsets $A, B$ of the plane are betweenness isomorphic if there is a bijection $f\colon A\to B$ such that, for every $x,y,z\in A$, the point $f(z)$ lies on the line segment connecting $f(x)$ and $f(y)$ if and only if $z$ lies on the line segment connecting $x$ and $y$. In general, it is quite difficult to tell whether two given subsets of the plane are betweenness isomorphic. We concentrate on the case when the sets $A,B$ belong to the family $ \mathcal A_c$ of unions of pairs of concentric circles in the plane. We prove that $A, B \in \mathcal A_c$ are betweenness isomorphic if and only if they are similar. In particular, there are continuum many betweenness isomorphism classes in $ \mathcal A_c$, and each of these classes consists exactly of all scaled translations of an arbitrary representative of the class. Furthermore, we show that every betweenness isomorphism between sets $A,B\in \mathcal A_c$ is exactly the restriction of a scaled isometry of the plane.