The volume of an isocanted cube is a determinant

TL;DR

This paper derives exact volume formulas for isocanted cubes and their polars in any dimension, confirming the Mahler conjecture for these specific convex bodies using elementary methods.

Contribution

It provides explicit volume formulas for isocanted cubes and their polars, and proves the Mahler conjecture for this class of origin-symmetric zonotopes.

Findings

Volume of isocanted cube equals the determinant of a specific matrix.

Volume of polar dual is a rational function in parameters l and a.

Mahler conjecture holds for isocanted cubes.

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.