Minimizing Visible Edges in Polyhedra

Abstract

We prove that, given a polyhedron $\mathcal{P}$ in ${\mathbb{R}}^3$, every point in ${\mathbb{R}}^3$ that does not see any vertex of $\mathcal{P}$ must see eight or more edges of $\mathcal{P}$, and this bound is tight. More generally, this remains true if $\mathcal{P}$ is any finite arrangement of internally disjoint polygons in ${\mathbb{R}}^3$. We also prove that every point in ${\mathbb{R}}^3$ can see six or more edges of $\mathcal{P}$ (possibly only the endpoints of some these edges) and every point in the interior of $\mathcal{P}$ can see a positive portion of at least six edges of $\mathcal{P}$. These bounds are also tight.

19 pages, 9 figures