Family of Equal-Sized $n$-Dimensional Polyhedra Satisfying Cavalieri's Principle

We prove the equality of the $(n-1)$-dimensional volumes of the cross-sections by parallel hyperplanes of a large family of $n$-dimensional convex polyhedra with nonnegative integer coordinates of their vertices, including the unit cube and the rectangular simplex with “legs” of lengths $1,2,\dots,n$. The cross-sections are perpendicular to the main diagonal of the cube. The first proof is carried out by a gradual reconstruction of the polyhedra, while the second one employs a direct calculation of the volumes by representing the polyhedra as the algebraic sum of convex cones.