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.