Scalar curvature can control the volume of geodesic ball locally, however, it can not bound the diameter. As forfar as I know, the example for a manifold with a large scalar curvature and volume has large diameter.
Comparing with the n sphere with the standard metric, if a smooth manifold has scalar curvature larger than this sphere, and the diameter of the manifold is smaller, should the volume of the manifold be smaller than the sphere?
