Abstract
We consider a variant of Heilbronn’s triangle problem by asking for fixed dimension d ≥ 2 and for fixed integers k ≥ 3 with k ≤ d+1 for a distribution of n points in the d-dimensional unit-cube [0,1]d such that the minimum volume of a k-point simplex among these n points is as large as possible. Denoting by Δ k,d (n) the supremum of the minimum volume of a k-point simplex among n points over all distributions of n points in [0,1]d we will show that c k . (log n)1/( d − − k + 2)/n (k − − 1)/(d − − k + 2) ≤ Δ k,d (n) ≤ c k ′/n (k − − 1)/d for 3 ≤ k ≤ d +1, and moreover Δ k,d (n) ≤ c k ′′/n (k − − 1)/d + (k − − 2)/(2d(d− − 1)) for k ≥ 4 even, and constants c k , c k ′, c k ′′ > 0.
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Lefmann, H. (2005). Distributions of Points in d Dimensions and Large k-Point Simplices. In: Wang, L. (eds) Computing and Combinatorics. COCOON 2005. Lecture Notes in Computer Science, vol 3595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11533719_52
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DOI: https://doi.org/10.1007/11533719_52
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28061-3
Online ISBN: 978-3-540-31806-4
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