Local Criteria in Polyhedral Minimizing Problems
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 $\mathbb{R}^4$ that are mass minimizing w.r.t. Lipschitz maps in the sense of Almgren's $M(0,\delta)$ sets as in Taylor's classification of two-dimensional soap film singularities. There are three that arise naturally by taking products of $\mathbb{R}$ 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 $2^{2/3}3^{11/6}\approx 11.896$. We present a variety of variational arguments to restrict the class of minimizing candidates.
Geometric Measure Theory, Minimizing Cones, Minimal Surfaces, Polyhedrons
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