Moser's Shadow Problem

Moser's shadow problem asks to estimate the shadow function $\mathfrak{s}_{b}(n)$ giving the minimal value with the property that for each bounded convex polyhedron $P$ in $3$-space with $n$ vertices there is some direction v (depending on P) such that when illuminated by parallel light rays from infinity in direction v the polyhedron casts a shadow having at least $\mathfrak{s}_{b}(n)$ vertices. A general version of the problem allows unbounded polyhedra as well, and has associated shadow function $\mathfrak{s}_{u}(n)$. This paper presents correct order of magnitude asymptotic bounds on these functions. The bounded case has answer $\mathfrak{b}(n) = \Theta \big( (\log n)/ (\log\log n)\big),$ a result following from 1989 work of Chazelle, Edelsbrunner and Guibas. The unbounded polyhedra case is shown to have the different asymptotic growth rate $\mathfrak{u}(n) = \Theta \big(1\big)$.