Generalizations of Heilbronn’s triangle problem

Abstract

For integers d, j ≥ 2 and n ≥ j, distributions of n points in the d-dimensional unit cube [0, 1] d are investigated, such that the minimum volume of the convex hull determined by j of n points is large. Lower and upper bounds on these minimum volumes are given. For obtaining lower bounds, results on the independence number of non-uniform, linear hypergraphs are used, which might be of interest by their own.

Upper bounds for Heilbronn's triangle problem in higher dimensions

“…For $$d= 2$$ the best lower bound is due to Lefmann [9]: $$$$\begin{equation} \Delta _{k, 2}(n) \gtrsim (\log n)^{\frac{1}{k-2}} n^{-\frac{k-1}{k-2}}, \end{equation}$$$$which is a logarithmic improvement over a bound by Bertram‐Kretzberg, Hofmeister, and Lefmann [3]. A simple probabilistic argument gives the following lower bound for arbitrary $$k > d \geqslant 2$$: $$$$\begin{equation} \Delta _{k, d}(n) \gtrsim n^{-\frac{k-1}{k-d}}, \end{equation}$$$$see [10] for a proof. It seems likely that one should be able to improve (6) by a logarithmic factor similarly to (5) but we do not pursue this direction here.…”

“…see [10] for a proof. It seems likely that one should be able to improve (6) by a logarithmic factor similarly to (5) but we do not pursue this direction here.…”