  For an asymptotically flat initial data, the Penrose inequality gives a lower
bound of the Arnowitt-Deser-Misner total mass of a spacetime in terms of the
area of certain surfaces representing black holes. This is a deep and beautiful
refinement of the famous positive mass theorem and it plays an important role
in the study of gravitational collapse. Gravitational collapse can also happen
if sufficient mass is concentrated into a finite region. This motivates us to
seek a localized version of the Penrose inequality. In this Letter, we
successfully make a precise statement of this form for the Liu-Yau quasi-local
mass in spherical symmetry.
