On the number of hinges defined by a point set in $\mathbb R^2$

TL;DR

This paper establishes a lower bound on the number of distinct three-point hinges in a planar point set, improving understanding of geometric configurations and extending incidence bounds in combinatorial geometry.

Contribution

It strengthens the Guth-Katz incidence estimate to a second moment bound and applies it to count hinges in the plane, advancing the analysis of geometric structures.

Findings

Number of three-point hinges is at least proportional to n^2 / log^3 n.

Extended Guth-Katz incidence bounds to a second moment estimate.

Connected hinge counts to the Erdős distinct distance problem.

Abstract

It is shown that the number of distinct types of three-point hinges, defined by a real plane set of $n$ points is $\gg n^2\log^{-3} n$, where a hinge is identified by fixing two pair-wise distances in a point triple. This is achieved via strengthening (modulo a $\log n$ factor) of the Guth-Katz estimate for the number of pair-wise intersections of lines in $\mathbb R^3$, arising in the context of the plane Erd\H os distinct distance problem, to a second moment incidence estimate. This relies, in particular, on the generalisation of the Guth-Katz incidence bound by Solomon and Sharir.