mg.metric geometry - Conic neighborhoods ⇔ Polyhedral

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 $Sigmatimes[0,r]$ with
  metric defined by the law of cosines; i.e.
  $$|(xi,x)(zeta,z)|^2=x^2+y^2-2xycosalpha,$$ where $alpha$ is the
  distance from $xi$ to $zeta$ in $Sigma$.