  In a Riemannian manifold, it is well known that the scalar curvature at a
point can be recovered from the volumes (areas) of small geodesic balls
(spheres). We show the scalar curvature is likewise determined by the relative
capacities of concentric small geodesic balls. This result has motivation from
general relativity (as a complement to a previous study by the author of the
capacity of large balls in an asymptotically flat manifold) and from weak
definitions of nonnegative scalar curvature. It also motivates a conjecture
(inspired by the famous volume conjecture of Gray and Vanhecke), regarding
whether Euclidean-like behavior of the relative capacity on the small scale is
sufficient to characterize a space as flat.
