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.