Thurston's Geometrization theorem implies that an acylindrical hyperbolic manifold M admits a unique hyperbolic metric whose convex core has totally geodesic boundary. We show that this is the most efficient hyperbolic metric on M, in the sense that it is the hyperbolic metric whose convex core has least possible volume.

The result above follows from an extension of work of Besson, Courtois and Gallot into the setting of Alexandrov spaces with lower bounds on curvature. This extension also has implications for volumes of hyperbolic cone manifolds.