Helly-type theorems for monotone properties of boxes

TL;DR

This work provides a unified combinatorial framework to derive Helly-type theorems for monotone properties of axis-aligned boxes, with corollaries for volume bounds and discrete containment. The authors introduce a general lemma showing that if every $2d$-tuple (or a fractional analogue on $d+1$-tuples) has property $P$, then the whole family has property $P$, and they extend this to $H$-convex sets via multiple orderings and supersaturation arguments. They obtain Colourful Helly-type statements and $(p,q)$-type results whose parameters depend only on the number of defining halfspaces $k$, not on the ambient dimension $d$, thus achieving a dimension-free framework. The results subsume and extend several existing Helly-type results for boxes and provide systematic proofs for a range of monotone properties.

Abstract

We present a unified approach to prove Helly-type theorems for monotone properties of boxes, such as having large volume or containing points from a given set. As a corollary, we obtain new proofs for several earlier results regarding specific monotone properties. Our results generalise to $H$-convex sets as well.

Theorems & Definitions (24)

• Lemma 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 2.1
• proof : Proof of Theorem \ref{['thm:HcolourfulStrongHelly']}