......Let e,d
Let $e$ and $d$ be real numbers such that $0 < e < d$. Are there known functions $B(e,d)$ that are upper bounds (close to or even equal to least upper bounds) for the surface area of the boundaries of (not necessarily convex) polyhedra in $E^3$ which have a diameter not greater than $d$ and every distinct pair of whose vertices have a distance apart not less than $e$? It is easy to construct examples showing that if we keep $d$ fixed and allow $e$ to approach 0, then $B(e,d)$ approaches infinity. Questions like this arise in connection with some recent theories of physics in which space (and perhaps also time) is "quantized". There is a minimum length $e$. Furthermore, the maximum amount of information that can be contained in any bounded region of space is limited. This limit is proportional, not to the volume of the region, but to the surface area of its boundary. One final question: Are there any simple necessary and sufficient conditions on a finite set of points $S$ in $E^3$ for there to exist a (convex or non-convex) polyhedron whose set of vertices is identical to $S$?
