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 1-norm, commonly referred to as the taxicab metric. Specifically, we investigate the following question: given d,k , what is the maximum size of a subset of 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
Mathematical Subject Classification 2010
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