Almost similar configurations

Abstract

Let $h(n)$ denote the maximum number of triangles with angles between $59^{\circ }$ and $61^{\circ }$ in any $n$-element planar set. Our main result is an exact formula for $h(n)$. We also prove $h(n)=n^3/24+O(n\log n)$ as $n\to \infty$. However, there are triangles $T$ and $n$-point sets $P$ showing that the number of $\epsilon$-similar copies of $T$ in $P$ can exceed $n^3/15$ for any $\epsilon >0$.

24 pages, 11 figures (A slightly revised version, a bit more detailed calculations)