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.