Large simplices determined by finite point sets (Q1943341)

For a finite set \(P\) in \(\mathbb{R}^n\) let \(d_1>d_2>\dotsb\) be all the distances spanned by \(P\) from the largest to the smallest. Previously, \textit{Z. Schur} et al. [Algorithms Comb. 25, 767--787 (2003; Zbl 1081.52016)] showed that there is at most one \(n\)-dimensional regular simplex with side length \(d_1\). In this paper, the result is extended: it is proved that the number of \(n\)-dimensional regular simplices with side length \(d_k\) is bounded by a function of \(n\) and \(k\) alone. The obtained bound is a tower of exponential of height \(\Omega(n)\). In addition, it is shown that in \(\mathbb{R}^3\) the distance \(d_k\) is realized at most \(c_k n\) times.