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

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.