Let K and L be compact convex sets in Rn. The following two statements are shown to be equivalent: (i) For every polytope Q ⊆ 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 ⊆ K having at most d+ 1 vertices, L contains a translate of Q. (i...

Every sequence of positive or negative homothetic copies of a planar convex body C whose total area does not exceed 0.175 times the area of C can be translatively packed in C. Let C be a planar convex body with area |C|. Moreover, let (Ci) be a finite or infinite sequence of homothetic copies of C. We say that (Ci) can be translatively packed in C if there exist translations σi such that σiCi a...

We consider information flow via physical transport of inscribed media through space and compare it to information flow via electromagnetic radiation. Somewhat counterintuitively for point to point links, physical transport of inscribed mass is often energetically more efficient by many orders of magnitude than electromagnetic broadcast. And perhaps more surprising, even in a broadcast setting ...