Euclidean d-simplices with \(k\)-faces of the same volume (Q1841874)

Inspired by the well-known fact that a simplex in \(\mathbb{R}^3\) with equal 2-face areas has congruent 2-faces and by a corresponding question of H. Lenz referring to higher dimensions, the author proves the following theorem: For \(d\geq 4\) and each integer \(k\) with \(3\leq k\leq d-1\), there is a Euclidean \(d\)-dimensional simplex all whose \(k\)-faces have equal \(k\)-volumes but are not congruent to each other. The elegant proof is mainly based on linear algebra, using e.g. eigenvectors and eigenvalues of matrices corresponding to the simplices.