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Fractional Helly theorem for Cartesian products of convex sets

Debsoumya Chakraborti, Jaehoon Kim, Jinha Kim, Minki Kim, Hong Liu

Abstract

Helly's theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly theorem for axis-aligned boxes, which are Cartesian products of line segments. Answering a question raised by Bárány and Kalai, and independently Lew, we generalize Eckhoff's result to Cartesian products of convex sets in all dimensions. In particular, we prove that given $α\in (1-\frac{1}{t^d},1]$ and a finite family $\mathcal{F}$ of Cartesian products of convex sets $\prod_{i\in[t]}A_i$ in $\mathbb{R}^{td}$ with $A_i\subset \mathbb{R}^d$ if at least $α$-fraction of the $(d+1)$-tuples in $\mathcal{F}$ are intersecting then at least $(1-(t^d(1-α))^{1/(d+1)})$-fraction of sets in $\mathcal{F}$ are intersecting. This is a special case of a more general result on intersections of $d$-Leray complexes. We also provide a construction showing that our result on $d$-Leray complexes is optimal. Interestingly the extremal example is representable as a family of cartesian products of convex sets, implying the bound $α>1-\frac{1}{t^d}$ and the fraction $(1-(t^d(1-α))^{1/(d+1)})$ above are also best possible. The well-known optimal construction for fractional Helly theorem for convex sets in $\mathbb{R}^d$ does not have $(p,d+1)$-condition for sublinear $p$. Inspired by this we give constructions showing that, somewhat surprisingly, imposing additional $(p,d+1)$-condition has negligible effect on improving the quantitative bounds in neither the fractional Helly theorem for convex sets nor Cartesian products of convex sets. Our constructions offer a rich family of distinct extremal configurations for fractional Helly theorem, implying in a sense that the optimal bound is stable.

Fractional Helly theorem for Cartesian products of convex sets

Abstract

Helly's theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly theorem for axis-aligned boxes, which are Cartesian products of line segments. Answering a question raised by Bárány and Kalai, and independently Lew, we generalize Eckhoff's result to Cartesian products of convex sets in all dimensions. In particular, we prove that given and a finite family of Cartesian products of convex sets in with if at least -fraction of the -tuples in are intersecting then at least -fraction of sets in are intersecting. This is a special case of a more general result on intersections of -Leray complexes. We also provide a construction showing that our result on -Leray complexes is optimal. Interestingly the extremal example is representable as a family of cartesian products of convex sets, implying the bound and the fraction above are also best possible. The well-known optimal construction for fractional Helly theorem for convex sets in does not have -condition for sublinear . Inspired by this we give constructions showing that, somewhat surprisingly, imposing additional -condition has negligible effect on improving the quantitative bounds in neither the fractional Helly theorem for convex sets nor Cartesian products of convex sets. Our constructions offer a rich family of distinct extremal configurations for fractional Helly theorem, implying in a sense that the optimal bound is stable.

Paper Structure

This paper contains 14 sections, 15 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Cartesian product of convex sets
  3. Intersection of $d$-Leray complexes
  4. Rich family of extremal configurations
  5. Intersection of $d$-Leray complexes
  6. No large subfamily without intersecting $(d+1)$-tuples
  7. Extremal example of convex sets in $\mathbb{R}^d$
  8. Extremal example of Cartesian products of convex sets in $\mathbb{R}^d$
  9. Concluding remarks
  10. Induced density vs forbidden blowup
  11. Fractional Helly properties of abstract set-systems
  12. The number of higher dimensional faces in $\mathcal{F}_{t,d}$
  13. Complexes with smaller dimensions
  14. $(p,q)$-condition for constant $p$

Key Result

Theorem 1.1

Let $d,r$, and $n$ be positive integers such that $n>d+r$, and $\mathcal{F}$ is a family of $n$ convex sets in $\mathbb{R}^d$. If more than $\binom{n}{d+1}-\binom{n-r}{d+1}$ of the $(d+1)$-tuples of the family $\mathcal{F}$ are intersecting, then $\mathcal{F}$ contains an intersecting subfamily of s

Figures (2)

  • Figure 1: If $A=A_{k+1}$ then conditions (ii), (iii), and (iv) hold.
  • Figure 2: $A_5$ is obtained by modifying $A$. It satisfies (i), (ii), (iii), and (iv).

Theorems & Definitions (29)

  • Theorem 1.1: The fractional Helly Theorem
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 2.1
  • Theorem 2.2: Turán's theorem, Tur41
  • Theorem \ref{thm:leray_intersection}
  • ...and 19 more