#### 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
##### Milestones
Received: 5 December 2019
Revised: 14 May 2020
Accepted: 23 May 2020
Published: 14 July 2020

Communicated by Anant Godbole
##### Authors
 Vajresh Balaji Department of Mathematics Millsaps College Jackson, MS United States Olivia Edwards Department of Mathematics Millsaps College Jackson, MS United States Anne Marie Loftin Department of Mathematics Millsaps College Jackson, MS United States Solomon Mcharo Department of Mathematics Millsaps College Jackson, MS United States Lo Phillips Department of Mathematics Millsaps College Jackson, MS United States Alex Rice Department of Mathematics Millsaps College Jackson, MS United States Bineyam Tsegaye Department of Mathematics Millsaps College Jackson, MS United States