Equiprojective polyhedra

A convex polyhedron P is equiprojective if, for some k, the orthogonal projection (or “shadow”) of P in every direction, except those directions parallel to faces of P, is a k-gon. We address an open question posed by Shepherd and reported in Croft, Falconer, and Guy's “Unsolved Problems in Geometry”, by characterizing equiprojective polyhedra, and giving an O(nlogn)-time recognition algorithm.