On the realization space of the cube (Q6192227)

Summary: 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.