Helly-type theorems for the diameter

TL;DR

This paper extends Helly's theorem to ensure intersections of convex sets in $\,R^d$ have large diameters, introducing colorful, fractional, and $(p,q)$ versions, along with related variants of Tverberg's theorem and epsilon-nets.

Contribution

It develops new diameter-based Helly-type theorems, including fractional and $(p,q)$ versions, and explores their connections to Tverberg's theorem and epsilon-net variants.

Findings

Fractional and $(p,q)$ Helly-type theorems hold with diameter guarantees.

Variants of Tverberg's theorem and epsilon-nets are established for convex sets with diameter bounds.

The results extend classical Helly's theorem to diameter-focused settings.

Abstract

We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets in $R^d$ has a large diameter. This includes colourful, fractional and $(p,q)$ versions of Helly's theorem. In particular, the fractional and $(p,q)$ versions work with conditions where the corresponding Helly theorem does not. We also include variants of Tverberg's theorem, B\'ar\'any's point selection theorem and the existence of weak epsilon-nets for convex sets with diameter estimates.