Cubes and adjoints of cross-polytopes

Abstract

We describe a bijection between oriented cubes and adjoints of cross-polytopes. This correspondence is used to prove that the real affine cube is, up to reorientation in the same class, the unique oriented cube that is realizable. Moreover, its underlying matroid is, up to isomorphism, the unique combinatorial cube that is realizable over the reals.

23 pages, 4 figures