Characterizing optimal point sets determining one distinct triangle

TL;DR

This paper determines the maximum size of point sets in various dimensions that form exactly one distinct triangle, revealing that the optimal configurations are simple and generalize previous results in geometric combinatorics.

Contribution

It extends the understanding of point configurations with a single triangle type to higher dimensions, providing exact maximums and characterizations.

Findings

Maximum points in 2D is 4 for one triangle type

Maximum points in 3D is 4 for one triangle type

Maximum points in higher dimensions is d+1 for one triangle type

Abstract

In this paper we determine the maximum number of points in $\mathbb{R}^d$ which form exactly $t$ distinct triangles, where we restrict ourselves to the case of $t = 1$. We denote this quantity by $F_d(t)$. It was known from the work of Epstein et al. 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. This is an extension of a variant of the distinct distance problem put forward by Erd\H{o}s and Fishburn.