Can you cover your shadows? (Q1076344)

The authors prove that a subset X of the Euclidean d-space \({\mathbb{R}}^ d\) can cover its shadows in \({\mathbb{R}}^ d\), if every orthogonal projection of X onto a (d-1)-dimensional linear subspace of \({\mathbb{R}}^ d\) is contained in some congruent copy of X. Every 2-dimensional convex set \(C\subset {\mathbb{R}}^ d\) has this property; but for \(d\geq 4\) no (d-1)- polytope. The authors consider also closely related problems and end with an interesting collection of open problems.