Covering shadows with a smaller volume

AbstractFor n⩾2 a construction is given for convex bodies K and L in Rn such that the orthogonal projection Lu onto the subspace u⊥ contains a translate of Ku for every direction u, while the volumes of K and L satisfy Vn(K)>Vn(L).A more general construction is then given for n-dimensional convex bodies K and L such that each orthogonal projection Lξ onto a k-dimensional subspace ξ contains a translate of Kξ, while the mth intrinsic volumes of K and L satisfy Vm(K)>Vm(L) for all m>k.For each k=1,…,n, we then define the collection Cn,k to be the closure (under the Hausdorff topology) of all Blaschke combinations of suitably defined cylinder sets (prisms).It is subsequently shown that, if L∈Cn,k, and if the orthogonal projection Lξ contains a translate of Kξ for every k-dimensional subspace ξ of Rn, then Vn(K)⩽Vn(L).The families Cn,k, called k-cylinder bodies of Rn, form a strictly increasing chainCn,1⊂Cn,2⊂⋯⊂Cn,n−1⊂Cn,n, where Cn,1 is precisely the collection of centrally symmetric compact convex sets in Rn, while Cn,n is the collection of all compact convex sets in Rn. Members of each family Cn,k are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes, and members of Cn,k are shown to satisfy certain geometric inequalities. Related open questions are also posed