# 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*.

## Keywords

Incidences Perpendicular bisectors Distinct distances Energy bound## Mathematics Subject Classification

52C10 05D99## Notes

### Acknowledgements

I thank Brandon Hanson, Peter Hajnal, Oliver Roche-Newton, Adam Sheffer, and Frank de Zeeuw for many stimulating conversations on perpendicular bisectors and related questions. I thank Luca Ghidelli for pointing out an error in Lemma 7 in an earlier version. I thank the anonymous referees for numerous helpful comments on the writing and presentation of this paper.

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