  In Riemannian geometry, the Cheng's maximal diameter rigidity theorem says
that if a complete $n$-manifold $M$ of Ricci curvature,
$\operatorname{Ric}_M\ge (n-1)$, has the maximal diameter $\pi$, then $M$ is
isometric to the unit sphere $S^n_1$. The main result in this paper is a
quantitative maximal diameter rigidity: if $M$ satisfies that
$\operatorname{Ric}_M\ge n-1$, $\operatorname{diam}(M)\approx \pi$, and the
Riemannian universal cover of every metric ball in $M$ of a definite radius
satisfies a Riefenberg condition, then $M$ is diffeomorphic and bi-H\"older
close to $S^n_1$.
