#### Vol. 13, No. 3, 2020

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

### Vajresh Balaji, Olivia Edwards, Anne Marie Loftin, Solomon Mcharo, Lo Phillips, Alex Rice and Bineyam Tsegaye

Vol. 13 (2020), No. 3, 487–509
##### 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 taxicab metric. Specifically, we investigate the following question: given $d,k\in ℕ$, what is the maximum size of a subset of ${ℝ}^{d}$ 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.

##### Keywords
Erdős distance problem, taxicab metric, discrete geometry, geometric combinatorics
Primary: 52C10