Let $\mathcal{A}\_\Gamma$ be the space of convex (non-degenerate) Euclidean polyhedra with $1$-skeleton a certain polyhedral graph $\Gamma$. This space can be seen as a subset of $\mathcal{Gr}\_2(\mathbb{R}^3)^F$ where $F$ is the number of faces of $\Gamma$.


It is a well known fact (for example in Proposition 17 of [Deformations of hyperbolic convex polyhedra and cone-$3$-manifolds](https://arxiv.org/pdf/0903.4743.pdf) by Montcouquiol) that $\mathcal{A}\_\Gamma$ is a manifold.


**Question:** Is $\mathcal{A}\_\Gamma$ connected?


My gut tells me the answer is yes but I couldn't find an easy proof nor a reference.


**Bonus questions:** Is it simply connected? Contractible?



Is the space of Euclidean polyhedra with a fixed $1$-skeleton connected?
I believe your question asks about *isotopy*:



> 
> **Isotopy property**: A combinatorial structure (such as a combinatorial type of
> polytope) has the isotopy property if any two realizations with the same orientation can be deformed into each other by a continuous deformation that maintains
> the combinatorial type. Equivalently, the isotopy property holds for a combinatorial structure if and only if its realization space is connected.
> 
> 
> 


This definition is quoted from "Basic Properties of Convex Polytopes"
by Martin Henk, Jürgen Richter-Gebert, and Günter M. Ziegler, in
*Handbook of Discrete and Computational Geometry*, 2017.
[CRC link](https://www.routledge.com/Handbook-of-Discrete-and-Computational-Geometry-Third-Edition/Toth-ORourke-Goodman/p/book/9781498711395).


Although the isotopy property holds for $3$-polytopes (Steinitz's Theorem),
it was proved by Richter-Gebert that it fails
(quite badly) in dimension $4$.
This is his "Universality Theorem."


