Volume bounds for shadow covering

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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 Lᵤ onto the subspace u⊥ contains a translate of the corresponding projection Kᵤ for every direction u, while the volumes of K and L satisfy Vₙ(K) > Vₙ(L). It is subsequently shown that, if the orthogonal projection Lᵤ onto the subspace u⊥ contains a translate of Kᵤ for every direction u, then the set (n/(n-1))L contains a translate of K. If follows that Vₙ(K) <= (n/(n-1))ⁿ Vₙ(L). In particular, we derive a universal constant bound Vₙ(K) <= 2.942 Vₙ(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.