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. $$1\le sec(M) \le K$$