Generalizations of Schöbi's tetrahedral dissection (Q1017915)

Let \({\mathcal Q}_n=\{x\in \mathbb R^n : x= \sum_{i=1}^n \lambda_iv_i \; (1\geq \lambda_1\geq\dots\geq\lambda_n\geq 0)\}\) denote a simplex with the edge vectors \(v_i\). \({\mathcal Q}_n={\mathcal Q}_n(w)\) is called a Hill-simplex of the first type if \(v_iv_j\equiv w\lambda^2\) with \((1-n)^{-1}<w<1\) for \(1\leq i<j\leq n\) and \(v_i^2=\lambda^2\) for \(1\leq i\leq n\). It is well known that \({\mathcal Q}_n(w)\) is equidissectable with an \(n\)-cube, but the proofs are not constructive. \textit{P. Schöbi} [Elem. Math. 40, 85--97 (1985; Zbl 0583.51018)] gave a three-piece dissection of \({\mathcal Q}_3(w)\) into a triangular prism. The authors generalize that to a dissection of \({\mathcal Q}_n(w)\) into \(n\) pieces that can be reassembled to form a prism \(c{\mathcal Q}_{n-1}(\frac{1}{n-1}) \times I\), where \(I\) denotes an interval. The results have applications to source coding and to constant-weight binary codes.