Volume Bounds for Shadow Covering
نویسندگان
چکیده
For n ≥ 2 a construction is given for a large family of compact convex sets K and L in Rn 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. It 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.
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