Geodesic
Family of Equal-Sized $n$-Dimensional Polyhedra Satisfying Cavalieri's Principle
Matematičeskie zametki, Tome 97 (2015) no. 2, pp. 231-248

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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.
Keywords: $n$-dimensional polyhedron, Cavalieri's principle, abelian group, pyramid, cone, cube.
Mots-clés : multiset 
@article{MZM_2015_97_2_a5,
     author = {F. M. Malyshev},
     title = {Family of {Equal-Sized} $n${-Dimensional} {Polyhedra} {Satisfying} {Cavalieri's} {Principle}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {231--248},
     publisher = {mathdoc},
     volume = {97},
     number = {2},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2015_97_2_a5/}
}
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F. M. Malyshev. Family of Equal-Sized $n$-Dimensional Polyhedra Satisfying Cavalieri's Principle. Matematičeskie zametki, Tome 97 (2015) no. 2, pp. 231-248. http://geodesic.mathdoc.fr/item/MZM_2015_97_2_a5/