Containment and inscribed simplices

Let K and L be compact convex sets in R^n. The following two statements are shown to be equivalent: (i) For every polytope Q inside K having at most n+1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 <= d <= n-1. It is also shown that the following two statements are equivalent: (i) For every polytope Q inside K having at most d+1 vertices, L contains a translate of Q. (ii) For every d-dimensional subspace W, the orthogonal projection of the set L onto W contains a translate of the corresponding projection of the set K onto W. It is then shown that, if K is a compact convex set in R^n having at least d+2 exposed points, then there exists a compact convex set L such that every d-dimensional orthogonal projection of L contains a translate of the corresponding projection of K, while L does not contain a translate of K. In particular, such a convex body L exists whenever dim(K) > d

Covering shadows with a smaller volume

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. V_m(K) > V_m(L)) for all m > i. It is then shown that, for each i = 1, ..., n, there is a class of bodies C{n,i}, called i-cylinder bodies of R^n, 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 C{n,i} are shown to form a strictly increasing chain of subsets C{n,1} < C{n,2} < ... < C{n,n-1} < C{n,n}, where C{n,1} is precisely the collection of centrally symmetric compact convex sets in n-dimensional space, while C{n,n} is the collection of all compact convex sets in n-dimensional space. Members of each family C{n,i} are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes, and members of C{n,i} are shown to satisfy certain geometric inequalities. Related open questions are also posed.Comment: 21 page