Optimal Point Sets With Few Distinct Triangles

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dc.contributor.authorDepret-Guillaume, James Sergeen
dc.contributor.committeechairPalsson, Eyvindur Arien
dc.contributor.committeememberSenger, Steven M.en
dc.contributor.committeememberOrr, Daniel D.en
dc.contributor.departmentMathematicsen
dc.date.accessioned2019-07-12T08:01:18Zen
dc.date.available2019-07-12T08:01:18Zen
dc.date.issued2019-07-11en
dc.description.abstractIn this thesis we consider the maximum number of points in $mathbb{R}^d$ which form exactly $t$ distinct triangles, which we denote by $F_d(t)$. We determine the values of $F_d(1)$ for all $dgeq3$, as well as determining $F_3(2)$. It was known from the work of Epstein et al. cite{Epstein} that $F_2(1) = 4$. Here we show somewhat surprisingly that $F_3(1) = 4$ and $F_d(1) = d + 1$, whenever $d geq 3$, and characterize the optimal point configurations. We also show that $F_3(2) = 6$ and give one such optimal point configuration. This is a higher dimensional extension of a variant of the distinct distance problem put forward by ErdH{o}s and Fishburn cite{ErdosFishburn}.en
dc.description.abstractgeneralIn this thesis we consider the following question: Given a number of triangles, t, where each of these triangles are different, we ask what is the maximum number of points that can be placed in d-dimensional space, such that every triplets of these points form the vertices of only the t allowable triangles. We answer this for every dimension, d when the number of triangles is t = 1, as well as show that when t = 2 triangle are in d = 3-dimensional space. This set of questions rises from considering the work of Erd˝os and Fishburn in higher dimensional space [EF].en
dc.description.degreeMaster of Scienceen
dc.format.mediumETDen
dc.identifier.othervt_gsexam:20941en
dc.identifier.urihttp://hdl.handle.net/10919/91425en
dc.publisherVirginia Techen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectOne triangle problemen
dc.subjectErdos problemen
dc.subjectOptimal configurationsen
dc.subjectFinite point configurationsen
dc.titleOptimal Point Sets With Few Distinct Trianglesen
dc.typeThesisen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.levelmastersen
thesis.degree.nameMaster of Scienceen

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