Volume bounds for shadow covering

Abstract

For n >= 2 a construction is given for a large family of compact convex sets K and L in n-dimensional Euclidean space such that the orthogonal projection L_u onto the subspace u^\perp contains a translate of the corresponding projection K_u for every direction u, while the volumes of K and L satisfy V_n(K) > V_n(L). It is subsequently shown that, if the orthogonal projection L_u onto the subspace u^\perp contains a translate of K_u for every direction u, then the set (n/(n-1))L contains a translate of K. If follows that V_n(K) <= (n/(n-1))^n V_n(L). In particular, we derive a universal constant bound V_n(K) <= 2.942 V_n(L), independent of the dimension n of the ambient space. Related results are obtained for projections onto subspaces of some fixed intermediate co-dimension. Open questions and conjectures are also posed.