A lower bound for Heilbronn's triangle problem in \(d\) dimensions (Q2719164)

Heilbronn's triangle problem asks for the smallest number \({\mathcal H}(n)\) such that among any \(n\) points in the unit square there are three points spanning a triangle of area less than \({\mathcal H}(n)\). This is a famous open problem, with an \(\Omega({\log n\over n^2})\) lower bound and an \(O({1\over n^{1.142}})\) upper bound. This suggests an immediate generalization to \(d\) dimensions: the smallest number \({\mathcal H}_d(n)\) such that among any \(n\) points in the \(d\)-dimensional unit cube there are \(d+1\) points spanning a simplex of volume at most \({\mathcal H}_d(n)\). NEWLINENEWLINENEWLINEThe author gives a lower bound \({\mathcal H}_d(n)=\Omega({1\over n^d})\), which he obtains once by a probabilistic construction and once by the modular moment curve. This lower bound is already superseded by an \(\Omega( {\log n\over n^d})\)-improvement by \textit{H. Lefmann} [SIAM, 60-64 (2000; Zbl 0958.52011)]. No upper bounds seem to be known.