Optimal Point Sets With Few Distinct Triangles

TR Number

Date

2019-07-11

Journal Title

Journal ISSN

Volume Title

Publisher

Virginia Tech

Abstract

In this thesis we consider the maximum number of points in mathbbRd which form exactly t distinct triangles, which we denote by Fd(t). We determine the values of Fd(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 Fd(1)=d+1, whenever dgeq3, 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}.

Description

Keywords

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

Citation

Collections