  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.
