Isosceles trapezoids, perpendicular bisectors, and few distinct distances

Ben Lund, Rutgers University

March 10, 2015

Erdős famously conjectured that every set of $n$ points in the real plane determines $\Omega(n/\sqrt{\log n})$ distinct distances. He further asked if a set of points that determines $O(n/\sqrt{\log n})$ distances must have some special structure; in particular, is there a line that contains contains $n^\varepsilon$ points, for some $\varepsilon>0$?

I will discuss a new upper bound on the number of isosceles trapezoids determined by a set of $n$ points, no more than $k$ of which lie on any circle or line, and the application of this bound to the aforementioned question of Erdős. I will also mention a result and an open problem on the number of distinct perpendicular bisectors determined by a set of points.

Based on joint work with Adam Sheffer and Frank de Zeeuw.