Suppose $M$ is a 1-connected closed manifold with sectional curvature $\ge 1$. So the diameter $D$ of $M$ satisfies
$$
D \le \pi
$$
When equality holds $M$ is isometric to round sphere. In fact this rigidity holds under the assumption of $Ric \ge n-1$. Hence it is natural to ask what happens to almost extreme case. i.e. when $$D \ge \pi -\epsilon $$Under the Ricci assumption only, the manifold is not necessary sphere by a counter example of M. Andersen. However with extra assumption: sectional curvature is bounded from below, Perelman proved it is homeomorphic to a twist sphere. (Is Perelman the first one prove this?)


By Grove-Shiohama's Diameter Sphere Thereom, under the assumption sectional curvature $\ge 1$ and $D\ge \pi-\epsilon$, $M$ is a twist sphere. (It also follows from Perelman's theorem above)


*My question is: Is $M$ diffeomorphic to the standard sphere?* 


Either Perelman's proof or Grove-Shiohama's Diameter sphere theorem uses the 'soft' approach, i.e. It is not by convergence argument, nor a Lipschitz distance between $M$ and $S^n$ is derived.


Actually, I suspect that it is not Gromov-Hausdorff close to the round sphere, as one might round off two tips of $S^2/\mathbb Z\_p$, where $\mathbb Z\_p$ acts on $S^2$ by rotation along the $z$-axis.


What if one assume there is also an upper bound $K$ on sectional curvatures, i.e. $$1\le sec(M) \le K$$



Positively curved manifold with almost extreme diameter
I will answer the last question.
Namely, let me show that if $1\le \mathrm{sec}\,M \le K$ and $\mathrm{diam}\, M>\pi-\varepsilon$ for sufficiently small $\varepsilon>0$ then it has to be diffeomorphic to $\mathbb S^m$.


Asssume contrary, then there is a sequence of $m$-dimensional manifolds $M\_n$ such that $1\le \mathrm{sec}\,M\_n \le K$ and $\mathrm{diam}\, M\_n\to\pi$ as $n\to \infty$.


Let's pass to a converging sequence $M\_n\to A$.
Note that $A$ is $m$-dimensional and $1\le \mathrm{sec}\, A \le K$ in the sense of Alexandrov and $\mathrm{diam}\, A=\pi$. 
It follows that $A$ is isometric to $\mathbb S^m$


Finally note that $M\_n$ is diffeomorphic to $A$ for all large $n$;
the later was essentally proved by Cheeger in his thesis.


