Radial Points in the Plane

Abstract

A radial point for a finite set P in the plane is a point q ∈ P with the property that each line connecting q to a point of P passes through at least one other element of P. We prove a conjecture of Pinchasi, by showing that the number of radial points for a non-collinear n-element set P is O(n). We also present several extensions of this result, generalizing theorems of Beck, Szemerédi and Trotter, and Elekes on the structure of incidences between points and lines.