Depth of segments and circles through points enclosing many points: a note

TL;DR

This paper improves bounds on the number of points enclosed by circles through pairs of points in a set, using a novel approach that leverages results about j-facets in three-dimensional point sets.

Contribution

It introduces a new method based on 3D j-facets to establish a stronger lower bound for points enclosed by circles through pairs of points.

Findings

Established a lower bound of n/4.7 points inside and outside for circles through pairs of points.

Provided a simpler proof for a known bound using 3D geometric results.

Enhanced previous bounds on point enclosure in planar point sets.

Abstract

Neumann-Lara and Urrutia showed in 1985 that in any set of n points in the plane in general positionthere is always a pair of points such that any circle through them contains at least (n-2)/60 points. In a series of papers, this result was subsequently improved till n/4.7, which is currently the best known lower bound. In this paper we propose a new approach to the problem that allows us, by using known results about j-facets of sets of points in $R^3$, to give a simple proof of a somehow stronger result: there is always a pair of points such that any circle through them has, both inside and outside, at least n/4.7 points.