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

Authors:
Hanno Lefmann;Niels Schmitt
Affiliations:
-;-
Venue:
LATIN '02 Proceedings of the 5th Latin American Symposium on Theoretical Informatics
Year:
2002

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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/n2). This conjecture was disproved by Koml贸s, Pintz and Szemer茅di [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 驴(log n/n2). 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 驴(log n/n3).