Separating families of convex sets

TL;DR

The paper establishes bounds on the number of convex sets needed to separate points in the plane, showing both upper and lower bounds that depend on the number of points.

Contribution

It provides new theoretical bounds on separating points with convex sets, improving understanding of geometric separation complexity.

Findings

Upper bound of O(n log log n / log n) convex sets for separation

Lower bound of Ω(n / log n) convex sets for some point sets

Advances in geometric separation theory

Abstract

Two elements, $x$ and $y$, are separated by a set $S$ if it contains exactly one of $x$ and $y$. We prove that any set of $n$ points in general position in the plane can be separated by $O(n\log\log n/\log n)$ convex sets, and for some point sets $\Omega(n/\log n)$ convex sets are necessary.