On the realization space of the cube

TL;DR

This paper studies the realization space of d-dimensional cubes, proving their connectivity via transformations, and uses this to explore the structure of cubical polytopes and their f-vectors within geometric cones.

Contribution

It establishes the connectivity of cube realization spaces and introduces an analog of the connected sum for cubical polytopes, advancing understanding of their geometric and combinatorial properties.

Findings

Realization space of the cube is connected and contractible.

Rays spanned by f-vectors of cubical polytopes are dense in Adin's cone.

Connectivity extends to products of simplices.

Abstract

We consider the realization space of the $d$-dimensional cube, and show that any two realizations are connected by a finite sequence of projective transformations and normal transformations. We use this fact to define an analog of the connected sum construction for cubical $d$-polytopes, and apply this construction to certain cubical $d$-polytopes to conclude that the rays spanned by $f$-vectors of cubical $d$-polytopes are dense in Adin's cone. The connectivity result on cubes extends to any product of simplices, and further, it shows the respective realization spaces are contractible.