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Helly theorem for affine spaces without dimensions

Sutanoya Chakraborty, Arijit Ghosh, Soumi Nandi

TL;DR

The paper extends Helly-type results to piercing by k-flats in Euclidean space under an unboundedness framework defined via limiting directions. It proves a colorful Helly theorem for k-flats and derives a no-dimensional variant giving explicit distance bounds to a common k-flat, with corresponding colorful versions. The results rely on projections, the colorful Helly theorem, and the limiting-direction framework to handle unbounded convex families, and they establish the necessity of k-unboundedness and the tightness of the bounds. Together, these contributions generalize prior no-dimensional Helly theorems from points to higher-dimensional affine subspaces and clarify the role of diameter bounds. Practical impact lies in sharpening piercing/coverage criteria for convex sets by affine subspaces in high dimensions.

Abstract

We prove a no-dimensional Helly theorem for affine spaces and convex sets using the unboundedness framework of Aronov, Goodman, and Pollack (Computational Geometry, 2002). This generalizes the fundamental result of Adiprasito, Bárány, Mustafa, and Terpai on the no-dimensional Helly theorem for points and convex sets (Discrete & Computational Geometry, 2020). Additionally, we establish the optimality of our result.

Helly theorem for affine spaces without dimensions

TL;DR

The paper extends Helly-type results to piercing by k-flats in Euclidean space under an unboundedness framework defined via limiting directions. It proves a colorful Helly theorem for k-flats and derives a no-dimensional variant giving explicit distance bounds to a common k-flat, with corresponding colorful versions. The results rely on projections, the colorful Helly theorem, and the limiting-direction framework to handle unbounded convex families, and they establish the necessity of k-unboundedness and the tightness of the bounds. Together, these contributions generalize prior no-dimensional Helly theorems from points to higher-dimensional affine subspaces and clarify the role of diameter bounds. Practical impact lies in sharpening piercing/coverage criteria for convex sets by affine subspaces in high dimensions.

Abstract

We prove a no-dimensional Helly theorem for affine spaces and convex sets using the unboundedness framework of Aronov, Goodman, and Pollack (Computational Geometry, 2002). This generalizes the fundamental result of Adiprasito, Bárány, Mustafa, and Terpai on the no-dimensional Helly theorem for points and convex sets (Discrete & Computational Geometry, 2020). Additionally, we establish the optimality of our result.
Paper Structure (7 sections, 9 theorems, 21 equations, 2 figures)

This paper contains 7 sections, 9 theorems, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations.
  3. Our results
  4. Proofs of the claimed results
  5. Helly theorems for $k$-flats: colorful and no-dimensional
  6. Necessity of $k$-unboundedness in Theorem \ref{['impossibility1']}
  7. Tightness of Theorem \ref{['main-result']}

Key Result

Theorem 1

Let $\mathcal{F}$ be a collection of compact convex sets in $\mathbb{R}^d$, $b \in \mathbb{R}^{d}$, and $1 \leq r \leq d$. Suppose that for every choice of $r$ sets $C_1, C_2, \dots, C_r$ in $\mathcal{F}$, their intersection contains a point from $B(b,1)$. There exists a point $q \in \mathbb{R}^d$ s

Figures (2)

  • Figure 1: An example demonstrating the necessity of $k$-unboundedness condition. This figure has been taken from AronovGP2002.
  • Figure 2: An example demonstrating the tightness of the bound given in Theorem \ref{['impossibility2']}.

Theorems & Definitions (15)

  • Theorem 1: Adiprasito, Bárány, Mustafa, and Terpai AdiprasitoBMT2020
  • Theorem 2: Adiprasito, Bárány, Mustafa, and Terpai AdiprasitoBMT2020
  • Remark 3
  • Theorem 4: Colorful Helly's theorem for $k$-flats
  • Theorem 5
  • Theorem 6: No-dimensional Helly theorem for $k$-flats
  • Theorem 7: Colorful no-dimensional Helly theorem for $k$-flats
  • Theorem 8: On families not being $k$-unbounded
  • Theorem 9: Tightness of the bound in Theorem \ref{['main-result']}
  • Theorem 10: Lovász and Bárány Barany82: Colorful Helly Theorem
  • ...and 5 more