The volume of an isocanted cube is a determinant

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.