Large simplices determined by finite point sets

Abstract

Given a set P of n points in $$\mathbb R ^{d}$$ , let $${d}_{1}>d_{2}>\cdots$$ denote all distinct inter-point distances generated by point pairs in $$P$$ . It was shown by Schur, Martini, Perles, and Kupitz that there is at most one d-dimensional regular simplex of edge length $${d}_{1}$$ whose every vertex belongs to P. We extend this result by showing that for any k the number of d-dimensional regular simplices of edge length $${d}_{k}$$ generated by the points of P is bounded from above by a constant that depends only on d and