Optimal Point Sets With Few Distinct Triangles

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}.