# The Number of Congruent Simplices in a Point Set

Discrete & Computational Geometry, Aug 2002

For 1 ≤ k≤ d-1 , let f k (d) (n) be the maximum possible number of k-simplices spanned by a set of n points in ℝ d that are congruent to a given k-simplex. We prove that $f_2^{(3)} (n) = O(n^{5/3} 2^{O(\alpha ^2 (n))} )$, f 2 (4) (n) = O(n 2+ε), for any ε > 0, f 2 (5) (n) = Θ(n 7/3), and f 3 (4) (n) = O(n 20/9+ε), for any ε > 0. We also derive a recurrence to bound f k (d) (n) for arbitrary values of k and d, and use it to derive the bound f k (d) (n) = O(n d/2+ ε), for any ε > 0, for d ≤ 7 and k ≤ d − 2. Following Erdős and Purdy, we conjecture that this bound holds for larger values of d as well, and for k ≤ d − 2.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00454-002-0727-x.pdf

Agarwal, Sharir. The Number of Congruent Simplices in a Point Set, Discrete & Computational Geometry, 2002, 123-150, DOI: 10.1007/s00454-002-0727-x