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), 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 for details.

  • $\begingroup$ Great question! $\endgroup$ – Gil Kalai Nov 13 '13 at 16:05
  • $\begingroup$ (But it sounds too good to be true...) $\endgroup$ – Gil Kalai Jan 31 '16 at 20:41

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