On the realization space of the cube
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.