A variant of the Hadwiger-Debrunner (p,q)-problem in the plane

TL;DR

This paper extends the Hadwiger-Debrunner (p,q)-problem in the plane, providing new bounds on the size of transversals for families of convex bodies with specific intersection properties related to a convex curve.

Contribution

It introduces a novel bound for families satisfying a (p,2)-condition, generalizing the problem to higher dimensions with the moment curve.

Findings

Transversal size bounded by 1.75×10^9 for certain intersection conditions.

For (p,2)-condition, transversal size is O(p^8).

Generalization to higher dimensions using the moment curve.

Abstract

Let $X$ be a convex curve in the plane (say, the unit circle), and let $\mathcal S$ be a family of planar convex bodies, such that every two of them meet at a point of $X$. Then $\mathcal S$ has a transversal $N\subset\mathbb R^2$ of size at most $1.75\cdot 10^9$. Suppose instead that $\mathcal S$ only satisfies the following "$(p,2)$-condition": Among every $p$ elements of $\mathcal S$ there are two that meet at a common point of $X$. Then $\mathcal S$ has a transversal of size $O(p^8)$. For comparison, the best known bound for the Hadwiger--Debrunner $(p, q)$-problem in the plane, with $q=3$, is $O(p^6)$. Our result generalizes appropriately for $\mathbb R^d$ if $X\subset \mathbb R^d$ is, for example, the moment curve.