Sets in R-d determining k taxicab distances

We address an analog of a problem introduced by Erdos and Fishburn, itself an inverse formulation of the famous Erdos 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 is an element of 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.