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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
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Abstract |
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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 -norm, commonly referred to as the taxicab metric. Specifically, we investigate the following question: given , what is the maximum size of a subset of that determines at most distinct taxicab distances, and can all such optimal arrangements be classified? We completely resolve the question in dimension , as well as the case in dimension , and we also provide a full resolution in the general case under an additional hypothesis. |
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Keywords
Erdős distance problem, taxicab metric, discrete geometry,
geometric combinatorics
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Mathematical Subject Classification 2010
Primary: 52C10
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Milestones
Received: 5 December 2019
Revised: 14 May 2020
Accepted: 23 May 2020
Published: 14 July 2020
Communicated by Anant Godbole
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Authors |
| Vajresh Balaji | |
| Department of Mathematics Millsaps College Jackson, MS United States |
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| Olivia Edwards | |
| Department of Mathematics Millsaps College Jackson, MS United States |
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| Anne Marie Loftin | |
| Department of Mathematics Millsaps College Jackson, MS United States |
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| Solomon Mcharo | |
| Department of Mathematics Millsaps College Jackson, MS United States |
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| Lo Phillips | |
| Department of Mathematics Millsaps College Jackson, MS United States |
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| Alex Rice | |
| Department of Mathematics Millsaps College Jackson, MS United States |
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| Bineyam Tsegaye | |
| Department of Mathematics Millsaps College Jackson, MS United States |
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