Equilateral Dimension of the Rectilinear Space

Designs Codes and Cryptography (Impact Factor: 0.96). 10/2000; 21(1-3):149-164. DOI: 10.1023/A:1008391712305
Source: DBLP


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: P 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 k + 1 for k 4. 1

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    • "At present Kusner's conjecture is proved for n = 1, 2 and 3 in [5], and for n = 4 in [9]. In connection with these results it also worth mentioning that it can be shown that the size of every 1 -equilateral set in R n does not exceed h(2n − 1) + 1 [9]. Therefore it is interesting to investigate if h(n) is linear in n. "

    Elemente der Mathematik 01/2007; DOI:10.4171/EM/55
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    • "The assertion of Conjecture 1.1 is easy for n ≤ 2, and has been proved for n = 3 in [2] and for n = 4 in [5]. For large n, the best known upper bound is 2 n − 1, and the existing techniques supplied no nontrivial upper bound. "
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