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Minimizing Visible Edges in Polyhedra (2208.09702v3)
Published 20 Aug 2022 in cs.CG and cs.DM
Abstract: We prove that, given a polyhedron $\mathcal P$ in $\mathbb{R}3$, every point in $\mathbb R3$ 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.