2007
DOI: 10.1007/s00454-007-9041-y
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Distributions of Points in d Dimensions and Large k-Point Simplices

Abstract: Abstract. We consider a variant of Heilbronn's triangle problem by investigating for fixed dimension d ≥ 2 and for integers k ≥ 2 with k ≤ d distributions of n points in the d-dimensional unit cube [0, 1] d such that the minimum volume of the simplices, which are determined by (k + 1) of these n points, is as large as possible. Denoting by ∆ k,d (n) the supremum of the minimum volume of a (k + 1)-point simplex among n points over all distributions of n points in, and, moreover, for odd integers k ≥ 1 we show t… Show more

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Cited by 3 publications
(8 citation statements)
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“…Simple probabilistic and packing arguments imply that nk1dk+2Δk,d(n)nk1d,$$\begin{equation} n^{- \frac{k-1}{d-k+2} } \lesssim \Delta _{k, d}(n) \lesssim n^{- \frac{k-1}{d}}, \end{equation}$$for any fixed 1kd+1$1 \leqslant k \leqslant d+1$ (see also [1, 2]). The lower bound can be improved by a logarithmic factor using results on independence numbers of sparse hypergraphs, see [8] for details. Lefmann [8] improved the upper bound by a polynomial factor for all even k2$k \geqslant 2$: Δk,d(n)nk1dk22dfalse(d1false),$$\begin{equation} \Delta _{k, d}(n) \lesssim n^{- \frac{k-1}{d} - \frac{k-2}{2d(d-1)}}, \end{equation}$$earlier, Brass [4] proved this in the special case k=d+1$k=d+1$.…”
Section: Introductionmentioning
confidence: 99%
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“…Simple probabilistic and packing arguments imply that nk1dk+2Δk,d(n)nk1d,$$\begin{equation} n^{- \frac{k-1}{d-k+2} } \lesssim \Delta _{k, d}(n) \lesssim n^{- \frac{k-1}{d}}, \end{equation}$$for any fixed 1kd+1$1 \leqslant k \leqslant d+1$ (see also [1, 2]). The lower bound can be improved by a logarithmic factor using results on independence numbers of sparse hypergraphs, see [8] for details. Lefmann [8] improved the upper bound by a polynomial factor for all even k2$k \geqslant 2$: Δk,d(n)nk1dk22dfalse(d1false),$$\begin{equation} \Delta _{k, d}(n) \lesssim n^{- \frac{k-1}{d} - \frac{k-2}{2d(d-1)}}, \end{equation}$$earlier, Brass [4] proved this in the special case k=d+1$k=d+1$.…”
Section: Introductionmentioning
confidence: 99%
“…The lower bound can be improved by a logarithmic factor using results on independence numbers of sparse hypergraphs, see [8] for details. Lefmann [8] improved the upper bound by a polynomial factor for all even k2$k \geqslant 2$: Δk,d(n)nk1dk22dfalse(d1false),$$\begin{equation} \Delta _{k, d}(n) \lesssim n^{- \frac{k-1}{d} - \frac{k-2}{2d(d-1)}}, \end{equation}$$earlier, Brass [4] proved this in the special case k=d+1$k=d+1$. For d3$d\geqslant 3$ and odd k$k$, no improvements over (2) were known and the first interesting special case is d=3$d=3$, k=3$k=3$.…”
Section: Introductionmentioning
confidence: 99%
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“…For integers d, j, n ≥ 2, let ∆ j,d (n) be the supremum, over all distributions of n points in the unit cube [0, 1] d , of the minimum ((j − 1)-dimensional for j ≤ d + 1) volume of a j-point simplex among n points. For fixed 3 ≤ j ≤ d + 1, the currently best bounds are [9].…”
Section: Introductionmentioning
confidence: 99%
“…For integers d, j, n ≥ 2, let ∆ j,d (n) be the supremum, over all distributions of n points in the unit cube [9].…”
Section: Introductionmentioning
confidence: 99%