Entanglement entropy in field theories is usually intractably difficult to calculate except in special cases. Through holography, however, the entanglement entropy of certain states simply corresponds to the area of a an extremal surface, in parallel with the Bekenstein-Hawking formula for black hole entropy. Interestingly, the Bekenstein-Hawking entropy of black holes places a lower bound on the total mass of an asymptotically flat spacetime via so-called Penrose inequalities. Here I will show that such flat-space Penrose inequalities can be generalized to local inequalities in asymptotically anti-de Sitter spacetimes, yielding a bound on the entanglement entropy of arbitrary regions of holographic field theories in terms of a weighted local energy density. A key tool in this generalization is the inverse mean curvature flow, which I will introduce and generalize to obtain these inequalities. I'll briefly discuss quantum extensions, but focus primarily on open questions and issues that could be addressed by mathematicians.