A Refined Energy Bound for Distinct Perpendicular Bisectors

Abstract

Let ${\mathcal {P}}$ be a set of n points in the Euclidean plane. We prove that, for any $\varepsilon > 0$, either a single line or circle contains n/2 points of ${\mathcal {P}}$, or the number of distinct perpendicular bisectors determined by pairs of points in ${\mathcal {P}}$ is $\Omega (n^{52/35 - \varepsilon })$, where the constant implied by the $\Omega$ notation depends on $\varepsilon$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of ${\mathcal {P}}$, or the number of distinct perpendicular bisectors is $\Omega (n^2)$. The proof relies bounding the size of a carefully selected subset of the quadruples $(a,b,c,d) \in {\mathcal {P}}^4$ such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d.