MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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).


  • 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 $\Sigma$.
share|cite|improve this question
I guess in the definition of a cone you meant to say that $r\alpha$ is the distance from $\xi$ to $\zeta$ in $\Sigma$? – Ramsay Feb 2 '12 at 14:05

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.