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 the upper bound −1)) , where c k,d , c k,d , c k,d > 0 are constants.

Upper bounds for Heilbronn's triangle problem in higher dimensions

“…Simple probabilistic and packing arguments imply that $$$$\begin{equation} n^{- \frac{k-1}{d-k+2} } \lesssim \Delta _{k, d}(n) \lesssim n^{- \frac{k-1}{d}}, \end{equation}$$$$for any fixed $$1 \leqslant k \leqslant d+1$$ (see also [1, 2]). The lower bound can be improved by a logarithmic factor using results on independence numbers of sparse hypergraphs, see [8] for details. Lefmann [8] improved the upper bound by a polynomial factor for all even $$k \geqslant 2$$: $$$$\begin{equation} \Delta _{k, d}(n) \lesssim n^{- \frac{k-1}{d} - \frac{k-2}{2d(d-1)}}, \end{equation}$$$$earlier, Brass [4] proved this in the special case $$k=d+1$$.…”

“…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 $$k \geqslant 2$$: $$$$\begin{equation} \Delta _{k, d}(n) \lesssim n^{- \frac{k-1}{d} - \frac{k-2}{2d(d-1)}}, \end{equation}$$$$earlier, Brass [4] proved this in the special case $$k=d+1$$. For $$d\geqslant 3$$ and odd $$k$$, no improvements over (2) were known and the first interesting special case is $$d=3$$, $$k=3$$.…”

“…for any fixed 1 ⩽ 𝑘 ⩽ 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 𝑘 ⩾ 2:…”

Generalizations of Heilbronn’s triangle problem

“…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].…”

Generalizations of Heilbronn's Triangle Problem

“…For integers d, j, n ≥ 2, let ∆ j,d (n) be the supremum, over all distributions of n points in the unit cube [9].…”