Characterizing optimal point sets determining one distinct triangle

Abstract

We determine the maximum number of points in ${ℝ}^d$ which form exactly $t$ distinct triangles, where we restrict ourselves to the case of $t=1$. We denote this quantity by $F_d\left(t\right)$. It is known from the work of Epstein et al. (Integers 18 (2018), art. id. A16) that $F_2\left(1\right)=4$. Here we show somewhat surprisingly that $F_3\left(1\right)=4$ and $F_d\left(1\right)=d+1$, whenever $d\ge 3$, and characterize the optimal point configurations. This is an extension of a variant of the distinct distance problem put forward by Erdős and Fishburn (Discrete Math. 160:1-3 (1996), 115–125).