Browsing by Author "Valfells, Asgeir"
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Item Local Criteria in Polyhedral Minimizing Problems(2023-04-21) Valfells, Asgeir; Hardt, Robert MThis 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.