Local Criteria in Polyhedral Minimizing Problems
type: CreativeWork
description: 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.
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'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.
author: Valfells, Asgeir
language: eng
keywords: Geometric Measure Theory, Minimizing Cones, Minimal Surfaces, Polyhedrons
date published: 2023-04-21
date modified: 2026-07-24T04:10:28Z
provider: Rice University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Rice University

PropertyValue
property ID: Handle
value: https://hdl.handle.net/1911/115193