Asymptotically efficient triangulations of the \(d\)-cube (Q1422223)

A triangulation of a convex \(d\)-polytope \(P\) is a decomposition of \(P\) into \(d\)-simplices by using only vertices of \(P\) as vartices of these simplices which intersect face to face. (The authors do not necessarily demand the latter condition.) In this paper triangulations of products \(P\times Q\) with few simplices are investigated, where \(P\) and \(Q\) are polytopes of ``low'' and ``high'' dimension, respectively. These results can be applied to get (asymptotically) efficient triangulations of the \(d\)-cube \(I^d\). Using the representation \(I^d= I^k\times I^{d-k}\) for some small \(k\) and applying their method, the authors succeed in obtaining a triangulation of \(I^d\) with \(0\) \((0.816^dd!)\) simplices.