Quantitative Fractional Helly and $(p,q)$-Theorems

TL;DR

This paper develops quantitative versions of classical Helly-type theorems, focusing on bounding the volume of intersections rather than just their existence, advancing geometric combinatorics.

Contribution

It introduces novel quantitative versions of the Fractional Helly Theorem and the $(p,q)$-Theorem, extending their applicability to volume bounds.

Findings

Established a quantitative Fractional Helly Theorem with volume bounds

Formulated a quantitative $(p,q)$-Theorem for volume constraints

Enhanced understanding of intersection volume properties in convex geometry

Abstract

We consider quantitative versions of Helly-type questions, that is, instead of finding a point in the intersection, we bound the volume of the intersection. Our first main geometric result is a quantitative version of the Fractional Helly Theorem of Katchalski and Liu, the second one is a quantitative version of the $(p,q)$-Theorem of Alon and Kleitman.