The Quantitative Fractional Helly theorem

TL;DR

The paper advances quantitative extensions of Helly-type theorems by uniting the Fractional Helly framework with the Quantitative Volume theorem. It establishes a sharp (d+1)-tuple quantitative Helly result and leverages it to obtain a quantitative $(p,d+1)$-theorem and its diameter analogue, supported by a robust transversality framework based on quantitative fractional transversals and weak epsilon nets. The authors also prove a QFH result for $(2d)$-tuples and derive a quantitative $(p,q)$-theorem, with diameter-variant counterparts, using hypergraph Ramsey and supersaturation methods alongside ellipsoid approximations. Collectively, the work sharpens our understanding of how fractional intersection patterns enforce positive-volume intersections and finite transversal structures in convex geometry.

Abstract

Two celebrated extensions of Helly's theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of Bárány, Katchalski, and Pach (1982). Improving on several recent works, we prove an optimal combination of these two results. We show that given a family $\mathcal{F}$ of $n$ convex sets in $\mathbb{R}^d$ such that at least $α\binom{n}{d+1}$ of the $(d+1)$-tuples of $\mathcal{F}$ have an intersection of volume at least 1, then one can select $Ω_{d,α}(n)$ members of $\mathcal{F}$ whose intersection has volume at least $Ω_d(1)$. Furthermore, with the help of this theorem, we establish a quantitative version of the $(p,q)$ theorem of Alon and Kleitman. Let $p\geq q\geq d+1$ and let $\mathcal{F}$ be a finite family of convex sets in $\mathbb{R}^d$ such that among any $p$ elements of $\mathcal{F}$, there are $q$ that have an intersection of volume at least $1$. Then, we prove that there exists a family $T$ of $O_{p,q}(1)$ ellipsoids of volume $Ω_d(1)$ such that every member of $\mathcal{F}$ contains at least one element of $T$. Finally, we present extensions about the diameter version of the Quantitative Helly theoerm.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Quantitative $(p,d+1)$ theorem
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1: Colorful Helly theorem
• Lemma 2.2: Quantitative Helly theorem
• Lemma 2.3: Galicer, Merzbacher, and Pinasco GMP
• Lemma 2.4
• Lemma 2.5