Hadwiger and Helly-type theorems for disjoint unit spheres

TL;DR

This paper establishes Helly-type theorems for line transversals to disjoint unit spheres in Euclidean space, providing conditions under which a family of such spheres admits a common transversal.

Contribution

It introduces new Helly-type theorems for line transversals to disjoint unit spheres, extending classical geometric results to higher dimensions and specific configurations.

Findings

A family of at least 2d disjoint unit balls has a line transversal if all 2d-sized subfamilies do, respecting a certain order.

A family of at least 4d-1 disjoint unit balls has a line transversal if all (4d-1)-sized subfamilies do.

The results generalize Helly-type theorems to the setting of disjoint unit spheres in Euclidean space.

Abstract

We prove Helly-type theorems for line transversals to disjoint unit balls in $\R^{d}$. In particular, we show that a family of $n \geq 2d$ disjoint unit balls in $\R^d$ has a line transversal if, for some ordering $\prec$ of the balls, any subfamily of 2d balls admits a line transversal consistent with $\prec$. We also prove that a family of $n \geq 4d-1$ disjoint unit balls in $\R^d$ admits a line transversal if any subfamily of size $4d-1$ admits a transversal.