Maximal \(j\)-simplices in the real \(d\)-dimensional unit cube (Q1369731)

The authors investigate the maximal volume of a \(j\)-simplex spanned by \(j\) vertices of the unit cube and the origin in real \(d\)-dimensional space. \textit{M. Hudelson}, \textit{V. Klee} and \textit{D. Larman} [Linear Algebr. Appl. 241-243, 519-598 (1996; Zbl 0861.15004)] established an upper bound for that volume; for \(j\) odd and specific values of \(d\) that bound is attained, whereas for \(j\) even that bound is not attainable. In the present paper the above mentioned result is improved by giving an upper bound for \(j\) even which is attained for infinitely many values of \(d\). Moreover, the cases \(j=2\) and \(j=3\) are discussed in detail.