Equilateral dimension of the rectilinear space

It is conjectured that there exist at most 2k equidistant points in the k-d imensional rectilinear space. This conjecture has been verified for k less than or equal to 3; we show here its validity in dimension k = 4. We also d iscuss a number of related questions. For instance, what is the maximum num ber of equidistant points lying in the hyperplane: Sigma (k)(i=1) x(i) = 0? If this number would be equal to k, then the above conjecture would follow . We show, however, that this number is greater than or equal to k + 1 for k greater than or equal to 4.