On Heilbronn’s Problem in Higher Dimension

Heilbronn conjectured that given arbitrary 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 a nonconstructive argument of Komlós, Pintz and Szemerédi [10] who showed that for every n there is a configuration of n points in the unit square [0, 1]2 where all triangles have area at least Ω(log n/n 2). Considering a generalization of this problem to dimensions d≥3, Barequet [3] showed for every n the existence of n points in the d-dimensional unit cube [0, 1]d such that the minimum volume of every simplex spanned by any (d+1) of these n points is at least Ω(1/n d). We improve on this lower bound by a logarithmic factor Θ(log n).

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Correspondence to Hanno Lefmann.

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Lefmann, H. On Heilbronn’s Problem in Higher Dimension. Combinatorica 23, 669–680 (2003). https://doi.org/10.1007/s00493-003-0040-1

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Mathematics Subject Classification (2000):

  • 68W25
  • 68R05
  • 05C69