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  • Cited by 3
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    APFELBAUM, ROEL and SHARIR, MICHA 2011. Non-Degenerate Spheres in Three Dimensions. Combinatorics, Probability and Computing, Vol. 20, Issue. 04, p. 503.


    DUMITRESCU, ADRIAN and TÓTH, CSABA D. 2008. On the Number of Tetrahedra with Minimum, Unit, and Distinct Volumes in Three-Space. Combinatorics, Probability and Computing, Vol. 17, Issue. 02,


    Solymosi, József and Vu, Van H. 2008. Near optimal bounds for the Erdős distinct distances problem in high dimensions. Combinatorica, Vol. 28, Issue. 1, p. 113.


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Distinct Distances in Three and Higher Dimensions

  • BORIS ARONOV (a1), JÁNOS PACH (a2), MICHA SHARIR (a3) and GÁBOR TARDOS (a4)
  • DOI: http://dx.doi.org/10.1017/S0963548304006091
  • Published online: 28 April 2004
Abstract

Improving an old result of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl, we show that the number of distinct distances determined by a set $P$ of $n$ points in three-dimensional space is $\Omega(n^{77/141-\varepsilon})=\Omega(n^{0.546})$, for any $\varepsilon>0$. Moreover, there always exists a point $p\in P$ from which there are at least so many distinct distances to the remaining elements of $P$. The same result holds for points on the three-dimensional sphere. As a consequence, we obtain analogous results in higher dimensions.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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