Similar Simplices in a d-Dimensional Point Set

We consider the problem of bounding the maximum possible number fk,d(n) of ksimplices that are spanned by a set of n points in R d and are similar to a given simplex. We first show that f2,3(n) = O(n 13/6), and then tackle the general case, and show that fd−2,d(n) = O(n d−8/5) and 1 fd−1,d(n) = O ∗ (n d−72/55), for any d. Our technique extends to derive bounds for other values of k and d, and we illustrate this by showing that f2,5(n) = O(n 8/3)