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Large Triangles in the d-Dimensional Unit-Cube

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Computing and Combinatorics (COCOON 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3106))

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Abstract

We consider a variant of Heilbronn’s triangle problem by asking for fixed dimension d ≥ 2 for a distribution of n points in the d-dimensional unit-cube [0,1]d such that the minimum (2-dimensional) area of a triangle among these n points is maximal. Denoting this maximum value by Δ d off − − line(n) and Δ d on − − line(n) for the off-line and the on-line situation, respectively, we will show that c 1 · (log n)1/(d − − 1)/n 2/(d − − 1) ≤ Δ d off − − line(n) ≤ C 1/n 2/d and c 2/n 2/(d − − 1) ≤ Δ d on − − line(n) ≤ C 2/n 2/d for constants c 1, c 2, C 1, C 2 > 0 which depend on d only.

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Author information

Authors and Affiliations

  1. Fakultät für Informatik, TU Chemnitz, D-09107, Chemnitz, Germany

    Hanno Lefmann

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  1. Hanno Lefmann
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Editors and Affiliations

  1. Division of Computer Science, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea

    Kyung-Yong Chwa

  2. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada

    J. Ian J. Munro

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© 2004 Springer-Verlag Berlin Heidelberg

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Lefmann, H. (2004). Large Triangles in the d-Dimensional Unit-Cube. In: Chwa, KY., Munro, J.I.J. (eds) Computing and Combinatorics. COCOON 2004. Lecture Notes in Computer Science, vol 3106. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27798-9_7

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  • DOI: https://doi.org/10.1007/978-3-540-27798-9_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22856-1

  • Online ISBN: 978-3-540-27798-9

  • eBook Packages: Springer Book Archive

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