I have a question about some defitions : 
  Orbifold, Alexandrov space, limit of manifolds in Gromov-Hausdorff distance sense.
  
   Consider following example. 

   Let $r> 0$   

   $L_c = \{ (x cos \theta, x sin \theta, cx) |  0 \leq x$ and $0 \leq \theta < 2\pi \}$ 

   $S$ : $(z-\sqrt{2} r)^2 + x^2 + y^2 = r^2$ 

   $T$ : $ (z- \sqrt{2} R)^2 + x^2 + y^2 = R^2$ 


   If $R$ is sufficiently large, then we have a two dimensional sphere $U_c$ enclosed by
   $L_c$, $S$, and $T$.


   First notice the following. $lim_{r \rightarrow 0} U_c$ is an 
   orbifold for some $c$ 


   Question : Is any orbifold is a limit of manifolds in Gromov-Hausdorff sense ? 

   If this question is wide, we can restrict to the case of nonnegatively curved orbifolds.
   : Is a nonnegatively curved $n$-orbifold a limit of positively curved $n$-manifolds ? 

   Question 2: In the following paper, a space with curvature $ \geq k$ is defined.

   M. Gromov Y. Burago and G. Perelman, A.d. alexandrov spaces with curvature bounded
   below, Uspekhi Mat. Nauk 47 (2) (1992), 3–51.

  Is a $n$-dimensional space with curvature $ \geq k$, which is smooth except 
  finite points, is a limit of $n$-manifolds of 
  positive sectional curvature $\geq k$ ? I believe that this question is trivial and 
  it is true.    


  I do not think that all orbifolds or spaces with curvature $ \geq k $ are limits of 
  manifolds. 

  However I can not deny it. 

  Since ${\bf R}^3={\bf R}^4 /S^1 = lim_{k \rightarrow \infty} {\bf R}^4/{\bf Z}_k$, 
  orbifolds are different from spaces with curvature $ \geq k $. But they are 
  obtained from the sequences of manifolds. 

  MOTIVATION : Hsiang-Kleiner classified positively curved manifolds with $S^1$-action. 
  I want to extend this result to positively curved orbifolds with $S^1$-action. 

  If orbifold is a limit of manifolds then the problem is simple. 

  Accordingly I want to know the questions. Thank you for your attention.