It is a well-know result (Steinitz, 1922) that the realization space of 3-dimensional convex polytopes with fixed combinatorics is contractible. 
A proof of this theorem can be found for instance in the book "Realization Spaces of Polytopes" (Springer Verlag, 1996)  by Richter-Gerbert. 

In his 1994 paper "Spatial Polygons and Stable Configurations of Points in the Projective Line" (see [here](http://link.springer.com/chapter/10.1007/978-3-322-99342-7_8))  Klyachko asked if the same conclusion (contractibility) still holds if in addition to combinatorics we also fix areas of faces of the polytope. Is this problem still open?