Critical values of homology classes of loops and positive curvature

We study compact and simply-connected Riemannian manifolds with positive sectional curvature $K\ge 1.$ For a non-trivial homology class of lowest dimension in the space of loops based at a point $p$ or in the free loop space one can define a critical length ${\mathsf{c}\mathsf{r}\mathsf{l}}_p(M,g)$ resp. ${\sf crl}\left(M,g\right).$Then${\sf crl}_p\left(M,g\right)$ equals the length of a geodesic loop and $\mathsf{c}\mathsf{r}\mathsf{l}(M,g)$ equals the length of a closed geodesic. This is the idea of the proof of the existence of a closed geodesic of positive length presented by Birkhoff in case of a sphere and by Lusternik and Fet in the general case. It is the main result of the paper that the numbers ${\mathsf{c}\mathsf{r}\mathsf{l}}_p(M,g)$ resp. $\mathsf{c}\mathsf{r}\mathsf{l}(M,g)$ attain its maximal value $2\pi$ only for the round metric on the $n$-sphere. Under the additional assumption $K\le 4$ this result for ${\sf crl}\left(M,g\right)$ follows from results by Sugimoto in even dimensions and Ballmann, Thorbergsson and Ziller in odd dimensions.