Equilateral Dimension of the Rectilinear Space

Abstract

It is conjectured that there exist at most 2k equidistant points in the k-dimensional rectilinear space. This conjecture has been verified for k ≤ 3; we show here its validity in dimension k = 4. We also discuss a number of related questions. For instance, what is the maximum number of equidistant points lying in the hyperplane: Σki=1 xi = 0? If this number would be equal to k, then the above conjecture would follow. We show, however, that this number is ≥ k + 1 for k ≥ 4.