We show that for a manifold with non-negative curvature one obtains a collection of concave functions, special cases of which are the concavity of the length of a Jacobi field in dimension $2$, and the concavity of the volume in general. We use these functions to show that there are many cohomogeneity one manifolds which do not carry an analytic invariant metric with non-negative curvature. This implies in particular, that one of the candidates in Positively curved cohomogeneity one manifolds and 3-Sasakian geometry does not carry an invariant metric with positive curvature.
"Concavity and rigidity in non-negative curvature." J. Differential Geom. 97 (2) 349 - 375, June 2014. https://doi.org/10.4310/jdg/1405447808