Article

# Equilateral Dimension of the Rectilinear Space

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

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: 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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Available from: Monique Laurent
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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 âˆ’ 1) + 1 [9]. Therefore it is interesting to investigate if h(n) is linear in n. "
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Preview · Article · Jan 2007 · Elemente der Mathematik
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• "The assertion of Conjecture 1.1 is easy for n â‰¤ 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 âˆ’ 1, and the existing techniques supplied no nontrivial upper bound. "
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ABSTRACT: We show that for every odd integer p 1 there is an absolute positive constantcp, so that the maximum cardinality of a set of vectors in Rn such that the lp distance between any pair is precisely 1, is at most cp n log n. We prove some upper bounds for other lp norms as well.
Preview · Article · May 2003 · Geometric and Functional Analysis
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• "Kusner [39] conjectured that this is tight, i.e., that e(l n 1 ) = 2n for all n. For n â‰¤ 4 this is proved in [44]. For general n, the best known upper bound is e(l n 1 ) â‰¤ c 1 n log n for some absolute positive constant c 1 . "
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ABSTRACT: Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. One of the main reasons for this growth is the tight connection between Discrete Mathematics and Theoretical Computer Science, and the rapid development of the latter. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage, and often relies on deep, well developed tools. This is a survey of two of the main general techniques that played a crucial role in the development of modern combinatorics; algebraic methods and probabilistic methods. Both will be illustrated by examples, focusing on the basic ideas and the connection to other areas.
Preview · Article · Dec 2002