Moser's Shadow Problem

Moser’s shadow problem asks to estimate the shadow function sb(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 sb(n) vertices. A general version of the problem allows unbounded polyhedra as well, and has associated shadow function su(n). This paper presents correct order of magnitude asymptotic bounds on these functions. The bounded case has answer sb(n) = Θ ( (logn)/(log log n) ) , a result following from 1989 work of Chazelle, Edelsbrunner and Guibas. The unbounded polyhedra case is shown to have the different asymptotic growth rate su(n) = Θ ( 1 ) .