Covering shadows with a smaller volume

For n ≥ 2 a construction is given for convex bodies K and L in R n such that the orthogonal projection Ku can be translated inside Lu 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 the orthogonal projection Kξ can be translated inside Lξ for every k-dimensional subspace ξ of Rn, while the m-th intrinsic volumes of K and L satisfy Vm(K) > Vm(L) for all m > k. It is then shown that, for each k = 1, . . . , n, there is a class of bodies Cn,k such that, if L ∈ Cn,k and if the orthogonal projection Kξ can be translated into Lξ for every k-dimensional subspace ξ ofRn, then Vn(K) ≤ Vn(L). The families Cn,k, called k-cylinder bodies of Rn, form a strictly increasing chain Cn,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. Suppose that K and L are compact convex subsets of n-dimensional Euclidean space. For a given dimension 1 ≤ k < n, suppose that every k-dimensional orthogonal projection (shadow) of K can be translated inside the corresponding projection of L. Does it follow that K has smaller volume than L? In this article it is shown that the answer in general is no. It is then shown that the answer is yes if L is chosen from a suitable family of convex bodies that includes certain cylinders and other sets with a direct sum decomposition. 1