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

reasonablemetric 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 $\Sigma$.