  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.
