similar simplices in a d-dimensional point set
We consider the problem of bounding the maximum possible number f k,d (n) of k-simplices that are spanned by a set of n pointsin R d and are similar to a given simplex. We first show that f 2,3 (n) = O(n 13/6 ), and then tacklethe general case, and show that f d-2, d (n) = O(n d-8/5 ) and f d-1,d (n) = O * (n d-72/55 ), for any d.Our technique extends to derive bounds for other valuesof k and d, and we illustrate this by showing that f 2,5 (n)=O(n 8/3 ).
incidences repeated patterns combinatorial geometry point configurations points and hyperplanes