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A Deterministic Polynomial Time Algorithm for Heilbronn’s Problem in Dimension Three

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Abstract

Heilbronn conjectured that among any n points in the 2- dimensional unit square [0, 1]2, there must be three points which form a triangle of area at most O(1/n 2). This conjecture was disproved by Komlós, Pintz and Szemerédi [15] who showed that for every n there exists a configuration of n points in the unit square [0, 1]2 where all triangles have area at least ω(log n/n 2). Here we will consider a 3-dimensional analogue of this problem and we will give a deterministic polynomial time algorithm which finds n points in the unit cube [0, 1]3 such that the volume of every tetrahedron among these n points is at least ω(log n/n 3).

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Authors and Affiliations

  1. Fakultät für Informatik, Technische Universität Chemnitz, Straße der Nationen 62, D-09107, Chemnitz, Germany

    Hanno Lefmann

  2. Lehrstuhl Mathematik und Informatik, Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780, Bochum, Germany

    Niels Schmitt

Authors
  1. Hanno Lefmann
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  2. Niels Schmitt
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Editor information

Editors and Affiliations

  1. Compaq Cambridge Research Laboratory, One Cambridge Center, 02142-1612, Cambridge, MA, USA

    Sergio Rajsbaum

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Lefmann, H., Schmitt, N. (2002). A Deterministic Polynomial Time Algorithm for Heilbronn’s Problem in Dimension Three. In: Rajsbaum, S. (eds) LATIN 2002: Theoretical Informatics. LATIN 2002. Lecture Notes in Computer Science, vol 2286. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45995-2_19

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  • DOI: https://doi.org/10.1007/3-540-45995-2_19

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43400-9

  • Online ISBN: 978-3-540-45995-8

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