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arXiv · 1910.00633

Characterizing optimal point sets determining one distinct triangle

Hazel N. Brenner·James S. Depret-Guillaume·Eyvindur A. Palsson·Robert W. Stuckey
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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.

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Hazel N. Brenner, James S. Depret-Guillaume, Eyvindur A. Palsson, Robert W. Stuckey. 2019-10-01. Characterizing optimal point sets determining one distinct triangle. https://doi.org/10.2140/involve.2020.13.91

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