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Abstract  

We give a very short survey of the results on placing of points into the unit n-dimensional cube with mutual distances at least one. The main result is that into the 5-dimensional unit cube there can be placed no more than 40 points.

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unit cube , Geom. Dedicata , 67 ( 1997 ) 285 - 293 . [7] Lassak , M. , A survey of algorithms for on-line packing and

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Authors: Vojtech Bálint, Vojtech Bálint and Pavel Novotný

., Bálint V. jr. On the number of point at distance at least one in the unit cube, Geombinatorics Vol. 12, 2003, pp. 157–166. Bálint V. On the number of point at distance at least one in

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Abstract  

We present a very short survey of known results and many new estimates and results on the maximum number of points that can be chosen in the n-dimensional unit cube so that every distance between them is at least 1.

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Abstract

This paper generalizes earlier results on the behaviour of uniformly distributed sequences in the unit interval [0,1] to more general domains. We devote special attention to the most interesting special case [0,1]d. This will naturally lead to a problem in geometric probability theory, where we generalize results by Anderssen, Brent, Daley and Moran about random chord lengths in high-dimensional unit cubes, thereby answering a question by Bailey, Borwein and Crandall.

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Abstract  

Let AN to be N points in the unit cube in dimension d, and consider the discrepancy function

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_N (\vec x): = \sharp \left( {\mathcal{A}_N \cap \left[ {\vec 0,\vec x} \right)} \right) - N\left| {\left[ {\vec 0,\vec x} \right)} \right|$$ \end{document}
Here,
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\vec x = \left( {\vec x,...,x_d } \right),\left[ {0,\vec x} \right) = \prod\limits_{t = 1}^d {\left[ {0,x_t } \right),}$$ \end{document}
and
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left| {\left[ {0,\vec x} \right)} \right|$$ \end{document}
denotes the Lebesgue measure of the rectangle. We show that necessarily
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\| {D_N } \right\|_{L^1 (log L)^{(d - 2)/2} } \gtrsim \left( {log N} \right)^{\left( {d - 1} \right)/2} .$$ \end{document}
In dimension d = 2, the ‘log L’ term has power zero, which corresponds to a Theorem due to [11]. The power on log L in dimension d ≥ 3 appears to be new, and supports a well-known conjecture on the L1 norm of DN. Comments on the discrepancy function in Hardy space also support the conjecture.

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: local minimality of the unit cube, http://arxiv.org/pdf/0905.0867v1 , Duke Math. J. , 154 (2010), 419–430. MR 2012a :52010 Petty, C. M. , Affine isoperimetric problems, Ann. N.Y. Acad. Sci

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