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A refined energy bound for distinct perpendicular bisectors. (English) Zbl 1441.52015

Summary: 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 \(d\).

MSC:

52C10 Erdős problems and related topics of discrete geometry
05D99 Extremal combinatorics

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References:

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