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EDIT: The Cheeger's theorem says that if $M^n$ is a compact smooth Riemannian manifold such that the absolute value of its sectional curvature is less than $\kappa$, diameter at most $D$, and volume at least $v>0$ then its injectivity radius is at least $i(n,\kappa,D,v)$, where $i(n,\kappa,D,v)$ depends only on $n,\kappa,D,v$.

Question. Does there exist an explicit estimate on the function $i(n,\kappa,D,v)$ (or at least some explicit estimate which can be extracted from a proof if one studies it carefully)?

Remark. I do not really need an estimate of this function, rather I am curious to understand whether the known methods of proof of the theorem use compactness arguments which do not lead to any explicit bounds.

A reference to the above result of Cheeger would be appreciated.

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    $\begingroup$ As stated, the estimate you give is wrong. To fix it you would also need an upper bound on the sectional curvature. To see that the upper bound is necessary, consider the round sphere $\mathbb{S}^2$ quotiented by an rotation of angle $\pi$ (it looks like a rugby ball). It has 2 conical singularities which can be smoothed out, and if you smooth those out just a little, the volume, diameter and curvature will have the bounds you require while the injectivity radius will be as small as wanted. $\endgroup$
    – Thomas Richard
    Jan 8, 2017 at 7:53
  • $\begingroup$ @ThomasRichard: Rugby balls are round everywhere; it looks more like an American football. $\endgroup$
    – Ben McKay
    Jan 8, 2017 at 8:06
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    $\begingroup$ The answer is yes. The proof uses the Sturm comparison theorem, which gives explicit bounds. Two references for Cheeger's theorem are the ones I gave in an earlier question: mathoverflow.net/questions/258509/… . I also just stumbled onto this: library.msri.org/books/Book30/files/abresch.pdf . The proof of the lower bound on the length of a closed geodesic relies on a theorem of Heintze-Karcher, which again uses the Sturm comparison theorem. I recommend reading their paper. $\endgroup$
    – Deane Yang
    Jan 8, 2017 at 17:26

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