On finite quasi-isometric sets in \(\mathbb R^n\) (Q1401600)

Let \(X=\{p_1, \dots, p_r\}\) be a finite set of points in the real Euclidean space \({\mathbb R}^n\) and let \(p_ip_j\) denote the distance between \(p_i\) and \(p_j\). Define \(B(p_i)\), the box of \(p_i\) in \(X\), as the collection of distances \(p_ip_j\), \(j=1,\dots,r\), \(j\neq i\). Two (finite) sets in \({\mathbb R}^n\) are quasi-isometric if there is a bijection between them such that the corresponding points have equal boxes; and isometric if there is a bijection such that all the distances between corresponding points are equal. The author shows that there are quasi-isometric sets that are not isometric, but if two four-point sets are quasi-isometric, then they are isometric. She also shows: If \(X=\{p_1, \dots, p_r\}\) and \(Y=\{q_1, \dots, q_r\}\) are two sets on the unit circle, where the distance is the arc length, and if \(B(p_i)=B(q_i)\) for \(i=1,2\), then under some additional assumptions the sets \(X\) and \(Y\) are isometric.