Lines pinning lines

TL;DR

This paper investigates the minimal configurations of convex polytopes in three-dimensional space that pin a line as a transversal, establishing upper bounds on the number of polytopes needed and characterizing the configurations for disjoint polytopes.

Contribution

It provides the first bounds on the size of minimal pinning configurations and characterizes all such configurations for disjoint convex polytopes.

Findings

Any minimal pinning of a line by convex polytopes without coplanar faces has at most eight polytopes.

If the polytopes are disjoint, the minimal pinning size reduces to at most six.

Complete characterization of disjoint polytope configurations that minimally pin a line.

Abstract

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show that any minimal pinning of a line by convex polytopes such that no face of a polytope is coplanar with the line has size at most eight. If, in addition, the polytopes are disjoint, then it has size at most six. We completely characterize configurations of disjoint polytopes that form minimal pinnings of a line.