Covering shadows with a smaller volume

For n⩾2n⩾2 a construction is given for convex bodies K and L   in RnRn such that the orthogonal projection LuLu onto the subspace u⊥u⊥ contains a translate of KuKu for every direction u, while the volumes of K and L   satisfy Vn(K)>Vn(L)Vn(K)>Vn(L).A more general construction is then given for n-dimensional convex bodies K and L   such that each orthogonal projection LξLξ onto a k-dimensional subspace ξ   contains a translate of KξKξ, while the mth intrinsic volumes of K and L   satisfy Vm(K)>Vm(L)Vm(K)>Vm(L) for all m>km>k.For each k=1,…,nk=1,…,n, we then define the collection Cn,kCn,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,kL∈Cn,k, and if the orthogonal projection LξLξ contains a translate of KξKξ for every k-dimensional subspace ξ   of RnRn, then Vn(K)⩽Vn(L)Vn(K)⩽Vn(L).The families Cn,kCn,k, called k  -cylinder bodies of RnRn, form a strictly increasing chainCn,1⊂Cn,2⊂⋯⊂Cn,n−1⊂Cn,n,Cn,1⊂Cn,2⊂⋯⊂Cn,n−1⊂Cn,n, where Cn,1Cn,1 is precisely the collection of centrally symmetric compact convex sets in RnRn, while Cn,nCn,n is the collection of all compact convex sets in RnRn. Members of each family Cn,kCn,k are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes, and members of Cn,kCn,k are shown to satisfy certain geometric inequalities. Related open questions are also posed.