Minimizing edge-length polyhedrons

Abstract

A 1957 conjecture by Zdzislaw Melzak, that the unit volume polyhedron with least edge length was a triangular right prism, with edge length $2^{2/3}3^{11/6}$. We present a variety of necessary local criteria for any minimizer. In the case that we are restricted to convex polyhedrons we demonstrate that all vertices must be of degree three, the number of triangular faces is at most 14, and we describe the behavior of quadrilateral faces should they become arbitrarily small.