On the realization space of the cube

We prove that the realization space of the d -dimensional cube is contractible. For this we first show that any two realizations are connected by a finite sequence of projective transformations and normal transformations. As an application 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 that the respective realization spaces are contractible.