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

Abstract

We address an analog of a problem introduced by Erdős and Fishburn, itself an inverse formulation of the famous Erdős distance problem, in which the usual Euclidean distance is replaced with the metric induced by the ${\ell }^1$-norm, commonly referred to as the $\text{taxicab metric}$. 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.

16 pages, 1 figure with 2 images, 1 table. Typos corrected, sections 5 and 6 revised to be more detailed, structured, and readable. Version to appear in Involve, a Journal of Mathematics

Cited by in corpus (1)

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