Cubes and adjoints of cross-polytopes

TL;DR

This paper establishes a bijection between oriented cubes and adjoints of cross-polytopes, proving the uniqueness of the real affine cube and its underlying matroid among realizable combinatorial cubes.

Contribution

It introduces a novel correspondence between oriented cubes and cross-polytope adjoints, demonstrating the uniqueness of the realizable real affine cube and its matroid.

Findings

The real affine cube is the unique realizable oriented cube.

Its underlying matroid is the only realizable combinatorial cube over the reals.

A bijection between oriented cubes and cross-polytope adjoints is established.

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.