On the number of congruent simplices in a point
Pankaj K. Agarwal, Micha Sharir · 2001
We derive improved bounds on the number of k-dimensional simplices spa nned by a set of n points in $\reals^d$ that are congruent to a given $k$-simplex, for $k\le d-1$. Let $f_k^{(d)} (n)$ be the maximum number of $k$-simplices spanned by a set of $n$ points in $\reals^d$ that are congruent to a given $k$-simplex. We prove that $f_2^{(3)}(n) = O(n^{5/3}\cdot 2^{O(\alpha^2(n))})$, $f_2^{(4)} (n) = O(n^{2+\eps})$, $f_2^{(5)} (n) = \Theta(n^{7/3})$, and $f_3^{(4)} (n) = O(n^{9/4+\eps})$. 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 $f_k^{(d)} (n) = O(n^{d/2})$ for $d \le 7$ and $k \le d-2$. Following Erd{\H o}s and Purdy, we conjecture that this bound holds for larger values of $d$ as well, and for $k\le d-2$.