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-Gebert. 

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?

Addendum: The main motivation for this question is that in the case of positive answer one obtains a natural cell decomposition of the moduli space $M_r$ of polygons $P$ in $R^3$ with fixed side-lengths $r_1,...,r_n$ as follows. Each polygon $P$ is defined by the $n$-tuple of vectors $(e_1,...,e_n)$ which add up to zero; the vectors $e_i$ represent the edges of $P$. Given this data, one uses Minkowski's theorem to construct a unique convex polytope $D_P$ in $R^3$ whose faces are orthogonal to the vectors $e_i$ and have areas $r_i=|e_i|$. Then one subdivides $M_r$ according to the combinatorial type of $D_P$. One would like to claim that this is a cell decomposition. A similar construction works nicely in the case of planar polygons. In this context, the combinatorial type of $D_P$ is replaced by the cyclically ordered partition of the set $\{1,...,n\}$ defined by the "Gauss map" sending $P$ to the $n$-tuple of points in $S^1$ corresponding to the directions of edges of $P$. See e.g. [here](http://arxiv.org/pdf/1209.3241v4.pdf) for details.