Distinct distances in three and higher dimensions

Boris Aronov, János Pach, Micha Sharir, Gabor Tardos

    Research output: Contribution to journalArticle

    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 Ω(n 77/141-ε) = Ω(n 0.546), for any Ε > 0. Moreover, there always exists a point p ∈ 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.

    Original languageEnglish (US)
    Pages (from-to)283-293
    Number of pages11
    JournalCombinatorics Probability and Computing
    Volume13
    Issue number3
    DOIs
    StatePublished - May 2004

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    Higher Dimensions
    Three-dimension
    Distinct
    Three-dimensional

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Computational Theory and Mathematics
    • Mathematics(all)
    • Discrete Mathematics and Combinatorics
    • Statistics and Probability

    Cite this

    Distinct distances in three and higher dimensions. / Aronov, Boris; Pach, János; Sharir, Micha; Tardos, Gabor.

    In: Combinatorics Probability and Computing, Vol. 13, No. 3, 05.2004, p. 283-293.

    Research output: Contribution to journalArticle

    Aronov, Boris ; Pach, János ; Sharir, Micha ; Tardos, Gabor. / Distinct distances in three and higher dimensions. In: Combinatorics Probability and Computing. 2004 ; Vol. 13, No. 3. pp. 283-293.
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