Abstract
Itis conjectured that there exist at most 2k equidistantpoints in the k-dimensional rectilinear space.This conjecture has been verified for k ≤ 3; we show here its validity in dimension k = 4. We alsodiscuss a number of related questions. For instance, what isthe maximum number of equidistant points lying in the hyperplane:\(\sum\nolimits_{i = 1}^k {x_i } = 0?\) If this number would be equal to k, then the above conjecture would follow. Weshow, however, that this number is ≥ k + 1 for k ≥ 4.
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Koolen, J., Laurent, M. & Schrijver, A. Equilateral Dimension of the Rectilinear Space. Designs, Codes and Cryptography 21, 149–164 (2000). https://doi.org/10.1023/A:1008391712305
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DOI: https://doi.org/10.1023/A:1008391712305