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$k$-dimensional transversals for fat convex sets

Attila Jung, Dömötör Pálvölgyi

TL;DR

The paper establishes a fractional Helly-type theorem for $k$-transversals of $\rho$-fat convex sets in $\mathbb{R}^d$: for any $d$, $\rho$, and $\alpha\in(0,1]$, there is a function $\beta(d,\rho,\alpha)$ such that if at least $\alpha$ of the $(k+2)$-tuples have a $k$-transversal, a $k$-flat intersects at least a $\beta$-fraction of the sets. It extends the result to colorful and spherical settings and proves a $(p,k+2)$-theorem for $k$-flats intersecting balls, using a projection-based inductive approach that reduces to lower-dimensional spherical problems. The proofs combine fatness-preserving projections, spherical cap geometry, and the Alon–Kleitman framework (with Haussler–Welzl epsilon-net ideas) to derive finite piercing bounds and their corollaries. These results generalize classical fractional Helly theorems to $k$-transversals under fatness and connect to transversal theory and weak epsilon-net questions, while highlighting open directions for general $\rho$-fat families.

Abstract

We prove a fractional Helly theorem for $k$-flats intersecting fat convex sets. A family $\mathcal{F}$ of sets is said to be $ρ$-fat if every set in the family contains a ball and is contained in a ball such that the ratio of the radii of these balls is bounded by $ρ$. We prove that for every dimension $d$ and positive reals $ρ$ and $α$ there exists a positive $β=β(d,ρ, α)$ such that if $\mathcal{F}$ is a finite family of $ρ$-fat convex sets in $\mathbb{R}^d$ and an $α$-fraction of the $(k+2)$-size subfamilies from $\mathcal{F}$ can be hit by a $k$-flat, then there is a $k$-flat that intersects at least a $β$-fraction of the sets of $\mathcal{F}$. We prove spherical and colorful variants of the above results and prove a $(p,k+2)$-theorem for $k$-flats intersecting balls.

$k$-dimensional transversals for fat convex sets

TL;DR

The paper establishes a fractional Helly-type theorem for -transversals of -fat convex sets in : for any , , and , there is a function such that if at least of the -tuples have a -transversal, a -flat intersects at least a -fraction of the sets. It extends the result to colorful and spherical settings and proves a -theorem for -flats intersecting balls, using a projection-based inductive approach that reduces to lower-dimensional spherical problems. The proofs combine fatness-preserving projections, spherical cap geometry, and the Alon–Kleitman framework (with Haussler–Welzl epsilon-net ideas) to derive finite piercing bounds and their corollaries. These results generalize classical fractional Helly theorems to -transversals under fatness and connect to transversal theory and weak epsilon-net questions, while highlighting open directions for general -fat families.

Abstract

We prove a fractional Helly theorem for -flats intersecting fat convex sets. A family of sets is said to be -fat if every set in the family contains a ball and is contained in a ball such that the ratio of the radii of these balls is bounded by . We prove that for every dimension and positive reals and there exists a positive such that if is a finite family of -fat convex sets in and an -fraction of the -size subfamilies from can be hit by a -flat, then there is a -flat that intersects at least a -fraction of the sets of . We prove spherical and colorful variants of the above results and prove a -theorem for -flats intersecting balls.
Paper Structure (4 sections, 14 theorems, 1 figure)

This paper contains 4 sections, 14 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Fractional Helly Theorems
  3. Caps and projections on the sphere
  4. (p,k+2)-theorems

Key Result

Theorem 1

For every dimension $d$ there exists a function $\beta\colon (0,1] \to (0,1]$ with the following property. Let $\mathcal{F}$ be a finite family of convex sets in $\mathbb{R}^d$ and $\alpha \in (0,1]$. If at least $\alpha\binom{|\mathcal{F}|}{d+1}$ of the $(d+1)$-size subfamilies of $\mathcal{F}$ hav

Figures (1)

  • Figure 1: A property of fat convex sets

Theorems & Definitions (35)

  • Theorem 1: Katchalski and Liu katchalski1979problem
  • Theorem 2: Alon and Kalai alon1995bounding
  • Theorem 3: Alon, Kalai, Matoušek and Meshulam alon2002transversal
  • Theorem 4: Matoušek matousek2004bounded
  • Theorem 5
  • Theorem 6: Alon and Kleitman alon1992piercing
  • Conjecture 7
  • Theorem 8
  • Theorem 9: Bárány et al. barany2014colourful
  • Theorem 10
  • ...and 25 more