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For an asymptotically flat initial data, the Penrose inequality gives a lowerbound of the Arnowitt-Deser-Misner total mass of a spacetime in terms of thearea of certain surfaces representing black holes. This is a deep and beautifulrefinement of the famous positive mass theorem and it plays an important rolein the study of gravitational collapse. Gravitational collapse can also happenif sufficient mass is concentrated into a finite region. This motivates us toseek a localized version of the Penrose inequality. In this Letter, wesuccessfully make a precise statement of this form for the Liu-Yau quasi-localmass in spherical symmetry.

Localized Penrose inequality for the Liu-Yau mass in spherical symmetry