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arXiv · 1911.08067

Sets in $\mathbb{R}^d$ determining $k$ taxicab distances

Vajresh Balaji·Olivia Edwards·Anne Marie Loftin·Solomon Mcharo·Lo Phillips·Alex Rice·Bineyam Tsegaye
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Abstract

We address an analog of a problem introduced by Erd\H{o}s and Fishburn, itself an inverse formulation of the famous Erd\H{o}s distance problem, in which the usual Euclidean distance is replaced with the metric induced by the $\ell^1$-norm, commonly referred to as the $\textit{taxicab metric}$. Specifically, we investigate the following question: given $d,k\in \mathbb{N}$, what is the maximum size of a subset of $\mathbb{R}^d$ that determines at most $k$ distinct taxicab distances, and can all such optimal arrangements be classified? We completely resolve the question in dimension $d=2$, as well as the $k=1$ case in dimension $d=3$, and we also provide a full resolution in the general case under an additional hypothesis.

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Vajresh Balaji, Olivia Edwards, Anne Marie Loftin, Solomon Mcharo, Lo Phillips, Alex Rice, Bineyam Tsegaye. 2019-11-19. Sets in $\mathbb{R}^d$ determining $k$ taxicab distances. https://doi.org/10.2140/involve.2020.13.487

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