Covering shadows with a smaller volume
Abstract
For each i = 1, ..., n constructions are given for convex bodies K and L in n-dimensional Euclidean space such that each rank i orthogonal projection of K can be translated inside the corresponding projection of L, even though K has strictly larger m-th intrinsic volumes (i.e. Vm(K) > Vm(L)) for all m > i. It is then shown that, for each i = 1, ..., n, there is a class of bodies Cn,i, called i-cylinder bodies of Rn, such that, if the body L with i-dimensional covering shadows is an i-cylinder body, then K will have smaller n-volume than L. The families Cn,i are shown to form a strictly increasing chain of subsets Cn,1 < Cn,2 < ... < Cn,n-1 < Cn,n, where Cn,1 is precisely the collection of centrally symmetric compact convex sets in n-dimensional space, while Cn,n is the collection of all compact convex sets in n-dimensional space. Members of each family Cn,i are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes, and members of Cn,i are shown to satisfy certain geometric inequalities. Related open questions are also posed.