Sagitta, Lenses, and Maximal Volume

We give a characterization of critical points that allows us to define a metric invariant on all Riemannian manifolds $M$ with a lower sectional curvature bound and an upper radius bound. We show there is a uniform upper volume bound for all such manifolds with an upper bound on this invariant. We generalize results by Grove and Petersen and by Sill, Wilhelm, and the author by showing any such $M$ that has volume sufficiently close to this upper bound is diffeomorphic to the standard sphere $S^{n}$ or a standard lens space $S^n/\mathbb{Z}_m$ where $m\in\{2,3,\ldots\}$ is no larger than an a priori constant.