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arXiv · 2406.01565

The volume of an isocanted cube is a determinant

M. J de la Puente·P. L. Clavería
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Abstract

In any dimension d>=2, we give exact volume formulas of two mutually polar dual convex d--polytopes. The primal body is called isocanted cube of dimension d, depending on two real parameters 0<a<l. The limit case a=0 yields a d--cube of edge--length l. We prove that the volume of such a body is the determinant of the matrix of order d having diagonal entries equal to l and a elsewhere. We also compute the volume of the polar dual body, getting a rational expression in l and a, homogeneous of degree -d with rational coefficients. Isocanted cubes are origin--symmetric zonotopes. Zonoids (defined as the limits of families of zonotopes) satisfy the Mahler conjecture; in particular, zonotopes do. Nonetheless, we confirm (by elementary methods) that the Mahler conjecture holds for isocanted cubes.

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M. J de la Puente, P. L. Clavería. 2024-06-03. The volume of an isocanted cube is a determinant. https://arxiv.org/abs/2406.01565

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