Convex polygons and separation of convex sets

TL;DR

This paper proves a geometric property of disjoint convex sets in the plane, showing that a separating line for some pair also separates a large subcollection of the entire set.

Contribution

It introduces a novel separation result for convex sets, linking pairwise separation to large subcollections in the plane.

Findings

Existence of a pair of convex sets with a separating line that also separates a large subcollection.

Quantitative bound of at least n/18 sets in the subcollection.

Applicable to arrangements of convex sets in computational geometry.

Abstract

We prove that for any collection F of $n \ge 2$ pairwise disjoint compact convex sets in the plane there is a pair of sets A and B in F such that any line that separates A from B separates either A or B from a subcollection of F with at least n/18 sets.