Odd-distance and right-equidistant sets in the maximum and Manhattan metrics

Abstract

We solve two related extremal-geometric questions in the n−dimensional space R∞n equipped with the maximum metric. First, we prove that the maximum size of a right-equidistant sequence of points in R∞n equals 2n+1−1. A sequence is right-equidistant if each of the points is at the same distance from all the succeeding points. Second, we prove that the maximum number of points in R∞n with pairwise odd distances equals 2n. We also obtain partial results for both questions in the n-dimensional space R1n with the Manhattan distance.