In the first half of the talk, we introduce a new quasi-local mass with interesting properties along null flows off of a 2-sphere in spacetime or, equivalently, foliations of a null cone. We also show how certain, fairly generic, convexity assumptions on the null cone allows for a proof of the Penrose Conjecture. On the Black Hole Horizon, we find that the convexity assumptions become sharp; therefore, the second half of the talk will explore the existence of a class of Black Hole Horizons admitting such convexity. From this, building upon the work of S. Alexakis, we will show that the Schwarzschild Null Cone--the case of equality for the Penrose Conjecture--is also critical in light of recent work on the perturbation of stable, weakly isolated Horizons.