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.



Orbifold vs Alexandrove space vs Limit of manifolds
**Q1.** Note that one oriented orthonormal frame bundle $FO$ over a smooth orbifold $O$ is a smooth manifold.
This frame bundle admits a one-parameter family of metrics which collapse to the original manifold and its curvature can be made bounded from below.


*Proof.* Equip $FO$ with $SO(n)$-invariant metric, consider product space $[\varepsilon\cdot SO(n)]\times FO$ and factorize along diagonal action. (This often called Cheeger's trick, but I think it was known before Cheeger.) 


**Q2.** It is an open question. It is expected that cones over some positively curved manifolds can not be approximated. Vitali Kapovitch has examples of such $n$ dimesional cones which can not be approximated by $m$-dimensional manifolds with $m< n+8$.


