This thesis will discuss two polyhedral minimizing problems and the necessary local criteria we find any such minimizers must have. We will also briefly discuss an extension of a third minimizing problem to higher dimension.

The first result we present classifies the three-dimensional piecewise linear cones in R4 that are mass minimizing w.r.t. Lipschitz maps in the sense of Almgren's M(0,δ) sets as in Taylor's classification of two-dimensional soap film singularities. There are three that arise naturally by taking products of R with lower dimensional cases and earlier literature has demonstrated the existence of two with 0-dimensional singularities. We classify all possible candidates and demonstrate that there are no p.l. minimizers outside these five.

The second result we present is an assortment of criteria for edge-length minimizing polyhedrons. The aim is to get closer to answering a 1957 conjecture by Zdzislaw Melzak, that the unit volume polyhedron with least edge length was a triangular right prism, with edge length 22/3311/6≈11.896. We present a variety of variational arguments to restrict the class of minimizing candidates.