Large simplices determined by finite point sets

Given a set P of n points in ℝd, let d1 > d2 >...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 oned-dimensional regular simplex of edge length d1 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 dk generated by the points of P is bounded from above by a constant that depends only on d and k. © 2012 The Managing Editors