Local Criteria in Polyhedral Minimizing Problems

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dc.contributor.advisorHardt, Robert M
dc.creatorValfells, Asgeir
dc.date.accessioned2023-08-09T19:25:00Z
dc.date.available2023-08-09T19:25:00Z
dc.date.created2023-05
dc.date.issued2023-04-21
dc.date.submittedMay 2023
dc.date.updated2023-08-09T19:25:01Z
dc.description.abstractThis 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.
dc.format.mimetypeapplication/pdf
dc.identifier.citationValfells, Asgeir. "Local Criteria in Polyhedral Minimizing Problems." (2023) Diss., Rice University. <a href="https://hdl.handle.net/1911/115193">https://hdl.handle.net/1911/115193</a>.
dc.identifier.urihttps://hdl.handle.net/1911/115193
dc.language.isoeng
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.
dc.subjectGeometric Measure Theory
dc.subjectMinimizing Cones
dc.subjectMinimal Surfaces
dc.subjectPolyhedrons
dc.titleLocal Criteria in Polyhedral Minimizing Problems
dc.typeThesis
dc.type.materialText
thesis.degree.departmentMathematics
thesis.degree.disciplineNatural Sciences
thesis.degree.grantorRice University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy
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