{"id":{"repo_id":"rice","oai_identifier":"oai:repository.rice.edu:1911/115193"},"canonical_url":"https://search.dev.ndltd.org/etd/rice/oai:repository.rice.edu:1911/115193","repository":{"repo_id":"rice","name":"Rice University","base_url":"https://repository.rice.edu/server/oai/request"},"display":{"title":"Local Criteria in Polyhedral Minimizing Problems","abstract":"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&apos;s $M(0,\\delta)$ sets as in Taylor&apos;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.","abstract_html":"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 <span class=\"etd-inline-math\">\\mathbb{R}<sup>4</sup></span> that are mass minimizing w.r.t. Lipschitz maps in the sense of Almgren&amp;apos;s <span class=\"etd-inline-math\">M(0,&delta;)</span> sets as in Taylor&amp;apos;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 <span class=\"etd-inline-math\">2<sup>2/3</sup>3<sup>11/6</sup>\\approx 11.896</span>. We present a variety of variational arguments to restrict the class of minimizing candidates.","abstract_has_math":true,"creators":["Valfells, Asgeir"],"institution":"Rice University","degree_name":"Doctor of Philosophy","degree_level":"Doctoral","degree_discipline":"Natural Sciences","degree_department":null,"school":null,"contributors":[],"advisors":["Hardt, Robert M"],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-04-21","date_published":"2023-04-21","updated_at":"2026-07-24T04:10:28Z","subjects":["Geometric Measure Theory","Minimizing Cones","Minimal Surfaces","Polyhedrons"],"languages":["eng"],"rights":["Copyright 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."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1911/115193","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Hardt, Robert M"]},{"key":"dc:creator","label":"Author","values":["Valfells, Asgeir"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2023-08-09T19:25:00Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2023-08-09T19:25:00Z"]},{"key":"dc:date.issued","label":"Date","values":["2023-04-21"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Natural Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Rice University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Geometric Measure Theory","Minimizing Cones","Minimal Surfaces","Polyhedrons"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 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."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1911/115193"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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&apos;s $M(0,\\delta)$ sets as in Taylor&apos;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."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Local Criteria in Polyhedral Minimizing Problems"]}]}],"canonical_facts":{"dc:contributor.advisor":["Hardt, Robert M"],"dc:creator":["Valfells, Asgeir"],"dc:date.accessioned":["2023-08-09T19:25:00Z"],"dc:date.available":["2023-08-09T19:25:00Z"],"dc:date.issued":["2023-04-21"],"dc:description.abstract":["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&apos;s $M(0,\\delta)$ sets as in Taylor&apos;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.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/1911/115193"],"dc:language.iso":["eng"],"dc:rights":["Copyright 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:subject":["Geometric Measure Theory","Minimizing Cones","Minimal Surfaces","Polyhedrons"],"dc:title":["Local Criteria in Polyhedral Minimizing Problems"],"dc:type":["Thesis"],"thesis:degree_discipline":["Natural Sciences"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["Rice University"]},"updated_at":"2026-07-24T04:10:28Z"}