I am looking for a reference to the following fact (I can prove it my-self, but it should be known for a century).

Let $X$ be a reasonable metric space such that each point has a spherical neighborhood which is isometric to a cone. Then $X$ is a polyhedral space.

Reasonable means say compact plus finite Hausdorff dimension (I would be happy with anything which includes finite dimensional Alexandrov space).

Definitions:

• A finite simplicial complex $P$ with a metric is called polyhedral space if each simplex in $P$ is isometric to a flat simplex.
• A space $K$ is called cone if there is a metric space $\Sigma$ and $r>0$ such that $K$ is isometric to $\Sigma\times[0,r]$ with metric defined by the law of cosines; i.e. $$|(\xi,x)(\zeta,z)|^2=x^2+y^2-2xy\cos\alpha,$$ where $\alpha$ is the distance from $\xi$ to $\zeta$ in $\Sigma$.