The Topology of the set of line Transversals

TL;DR

This paper proves that the set of lines intersecting a collection of disjoint convex sets in three-dimensional space has a simple, contractible topology, extending to directed lines.

Contribution

It establishes the contractibility of the set of transversals for disjoint convex sets in 3D, a novel topological result in geometric transversal theory.

Findings

Connected components of line transversals are contractible.

Results hold for both lines and directed lines.

Advances understanding of the topology of geometric transversals.

Abstract

We prove that for any set $F$ of $n\ge 2$ pairwise disjoint open convex sets in $\mathbb{R}^3$, the connected components of the set of lines intersecting every member of $F$ are contractible. The same result holds for directed lines.