On unit bases for the Euclidean metric (Q1375023)

Let \((M,\rho )\) be a metric space. A set \(\mathcal B\subset M\times M\) is called a base of the metric \(\rho \) if each mapping \(f\:M\to M\) such that \(\rho (f(X),f(Y))=\rho (X,Y)\) for every \(\{ X,Y\}\in\mathcal B\) is an isometry. Due to a well-known theorem by \textit{F. S. Beckman} and \textit{D. A. Quarles} [Proc. Am. Math. Soc. 4, No. 5, 810-815 (1953; Zbl 0052.18204)], in the Euclidean \(n\)-dimensional space with a metric \(d\) the set of all pairs \(\{ X,Y\}\) of points satisfying \(d(X,Y)=1\) is a base of the Euclidean metric \(d\) for each \(n\geq 2\). In the article under review, it is in particular proven that, for \(n\geq 3\), there exists a base \(\widetilde{\mathcal B}\) of the Euclidean metric such that 1) \(d(X,Y)=1\) for each \(\{ X,Y\}\in\widetilde{\mathcal B}\) and 2) the segments \([X_1,Y_1]\) and \([X_2,Y_2]\) are not parallel for any distinct pairs \(\{X_i,Y_i\}\in\widetilde{\mathcal B}\) \((i=1,2)\). Validity of this statement for \(n=2\) still remains open.