  In this paper lower bounds are obtained for quasi-local masses in terms of
charge, angular momentum, and horizon area. In particular we treat three
quasi-local masses based on a Hamiltonian approach, namely the Brown-York,
Liu-Yau, and Wang-Yau masses. The geometric inequalities are motivated by
analogous results for the ADM mass. They may be interpreted as localized
versions of these inequalities, and are also closely tied to the conjectured
Bekenstein bounds for entropy of macroscopic bodies. In addition, we give a new
proof of the positivity property for the Wang-Yau mass which is used to remove
the spin condition in higher dimensions. Furthermore, we generalize a recent
result of Lu and Miao to obtain a localized version of the Penrose inequality
for the static Wang-Yau mass.
