The Spacetime Inequality of Penrose 1972 says that any slice in any black hole spacetime, the ADM mass of the slice is bounded below by the area of cuts with the event horizon. (Its quasi-local version is a refinement involving quasi-local notions of mass and energy). The Hoop Conjecture proposed by Thorne 1972 says that black holes form when the quasi-local mass M associated with some region of some characteristic length C satisfies ''M > const. C''. These conjectures are open in full generality and I will present some recent progress which proves them in a certain setting. This is joint with S.T. Yau and A. Alaee.