Optimal Point Sets With Few Distinct Triangles

In 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 F₃(2). It was known from the work of Epstein et al. cite{Epstein} that F₂(1) = 4. Here we show somewhat surprisingly that F₃(1) = 4 and F_d(1) = d + 1, whenever $d geq 3$, and characterize the optimal point configurations. We also show that F₃(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}.

Name thesis:degree_name

Master of Science

Level thesis:degree_level

masters

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2019

Author dc:creator

• Depret-Guillaume, James Serge

Chair dc:contributor.committeechair

• Palsson, Eyvindur Ari

Committee members dc:contributor.committeemember

• Senger, Steven M.
• Orr, Daniel D.

• One triangle problem
• Erdos problem
• Optimal configurations
• Finite point configurations

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Dc Identifier Other

vt_gsexam:20941

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/91425