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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
ℓ 1 -norm, commonly
referred to as the
taxicab metric . Specifically, we investigate the following question: given
d , k
∈
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
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