[Paper Review] Fractional Helly theorem for the diameter of convex sets

Silouanos Brazitikos|arXiv (Cornell University)|Nov 12, 2015

Point processes and geometric inequalities14 references3 citations

TL;DR

This paper establishes a new quantitative Helly-type theorem for the diameter of convex sets, showing that for any finite family of convex bodies in ℝⁿ with non-empty interior intersection, there exists a subfamily of O(n) sets whose intersection is contained in a scaled version of the full intersection—specifically, within O(n^{3/2}) times the original set. In the symmetric case, the bound improves to O(√n), achieving polynomial dependence on dimension and resolving a conjecture on diameter control in Helly-type settings.

ABSTRACT

We provide a new quantitative version of Helly's theorem: there exists an absolute constant $α>1$ with the following property: if $\{P_i: i\in I\}$ is a finite family of convex bodies in ${\mathbb R}^n$ with ${ m int}\left (\bigcap_{i\in I}P_i ight ) eq\emptyset $, then there exist $z\in {\mathbb R}^n$, $s\leq αn$ and $i_1,\ldots i_s\in I$ such that \begin{equation*} z+P_{i_1}\cap\cdots\cap P_{i_s}\subseteq cn^{3/2}\left(z+\bigcap_{i\in I}P_i ight), \end{equation*} where $c>0$ is an absolute constant. This directly gives a version of the "quantitative" diameter theorem of Bárány, Katchalski and Pach, with a polynomial dependence on the dimension. In the symmetric case the bound $O(n^{3/2})$ can be improved to $O(\sqrt{n})$.

Motivation & Objective

• To establish a quantitative version of Helly’s theorem for the diameter of convex sets, addressing whether 'large diameter' is a Helly-type property.
• To improve upon prior exponential bounds in diameter control by achieving polynomial dependence on dimension.
• To resolve a conjecture by Bárány, Katchalski, and Pach on whether the diameter of a subintersection can be bounded by O(√n) instead of O((cn)^{n/2}).
• To extend the framework of quantitative Helly theorems beyond volume to diameter, using geometric and spectral techniques.
• To provide explicit, dimension-dependent bounds for the diameter of intersections of convex sets under minimal intersection assumptions.

Proposed method

• Utilizes a lemma from Barvinok (2005) based on the method of interlacing families, which connects convex geometry with spectral graph theory.
• Applies a theorem of Batson, Spielman, and Srivastava (2012) on graph sparsifiers to control the geometry of finite point sets in ℝⁿ.
• Employs the concept of Löwner’s position for convex bodies and their polar bodies to reduce the problem to spectral control of positive definite matrices.
• Uses the John–Löwner ellipsoid and support function analysis to derive covering estimates for convex hulls of finite point sets.
• Applies a reduction scheme from [3] to further minimize the number of sets in the subfamily while preserving diameter bounds.
• Relies on the fact that contact points of the polar body with its minimal volume ellipsoid yield subfamilies with controlled intersection geometry.

Experimental results

Research questions

• RQ1 Can the diameter of a subintersection of convex sets be bounded by a polynomial in n, rather than an exponential or super-polynomial function?
• RQ2 Is the property 'the intersection has large diameter' a Helly-type property for convex sets, and if so, what is the optimal number of sets needed to control the diameter?
• RQ3 Can the O((cn)^{n/2}) bound from Bárány, Katchalski, and Pach be improved to O(√n) in the symmetric case?
• RQ4 What is the best possible dependence on dimension for the diameter of a subintersection when the full intersection has diameter 1?
• RQ5 Can the number of sets in the subfamily be reduced to O(n) while maintaining a polynomial diameter bound?

Key findings

• For any finite family of convex bodies in ℝⁿ with non-empty interior intersection, there exists a subfamily of size at most αn (for absolute α > 1) such that the intersection of these sets is contained in a set of diameter O(n^{3/2}) times the diameter of the full intersection.
• In the symmetric case, the bound improves to O(√n), and the constant depends explicitly on a parameter d > 1 as γ_d = (√d + 1)/(√d − 1), showing tightness.
• The result provides a polynomial dependence on dimension for the diameter of subintersections, improving upon the exponential bounds in prior work.
• The proof technique relies on spectral methods and convex geometry, particularly the use of the minimal volume ellipsoid and contact points of the polar body.
• The method yields a quantitative Helly theorem for diameter with explicit constants, and the bound is tight up to absolute constants.
• The result implies that the diameter control problem satisfies a Helly-type property with polynomial dependence on dimension, confirming a conjecture in qualitative form.