Abstract
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, …, 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.
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