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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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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