At the moment I am polishing my lecture notes which in particular cover Alexandrov's existence theorem.

Denote by $\mathbf{P}_k$ the space of isometry classes of polyhedral metrics on the $\mathbb S^2$ with k-verteces and nonnegative curvature at each vertex.

In the proof of Alexandrov's existence theorem one has to show that any space in $\mathbf{P}_k$ can be continuously deformed in $\mathbf{P}_k$ into a realizable polyhedral metric; i.e. the one which is isometric to the surface of convex polyhedron.

The original Alexandrov's proof is simple but tedious. First he deforms the given polyhedral space into space with one degenerate vertex. Second he shows that any polyhedral metric near the degenerate space is realizable.

It seems that I found a simpler way to do this. I proof that $\mathbf{P}_k$ is path connected by constructing a path between given spaces in $\mathbf{P}_k$ using induction on $k$. The existence of one realizable space in $\mathbf{P}_k$ is not a problem.

Was it done before?