Uniqueness of Optimal Point Sets Determining Two Distinct Triangles

TL;DR

This paper proves that in dimensions three and higher, the maximum number of points forming exactly two distinct triangles is 2d, uniquely achieved by the vertices of the d-orthoplex, with a simplified proof of this fact.

Contribution

It establishes the maximum point set size for exactly two triangle types in higher dimensions and proves the uniqueness of the d-orthoplex configuration, providing a more elementary proof.

Findings

Maximum of 2d points in d≥3 dimensions for exactly 2 triangle types

d-orthoplex uniquely achieves this maximum

Simplified proof of the orthoplex's optimality

Abstract

In this paper, we show that the maximum number of points in $d\geq3$ dimensions determining exactly 2 distinct triangles is $2d$. We further show that this maximum is uniquely achieved by the vertices of the $d$-orthoplex. We build upon the work of Hirasaka and Shinohara who determined that the $d$-orthoplex is such an optimal configuration, but did not prove its uniqueness. Further, we present a more elementary argument for its optimality.