  The Riemannian Penrose inequality is a remarkable geometric inequality
between the ADM mass of an asymptotically flat manifold with non-negative
scalar curvature and the area of its outermost minimal surface. A version of
the Riemannian Penrose inequality has also been established for the
Einstein-Maxwell equations, where the lower bound on the mass also depends on
the electric charge. In the context of quasi-local mass, one is interested in
determining if, and for which quasi-local mass definitions, a quasi-local
version of these inequalities also holds.
  It is known that the Brown-York quasi-local mass satisfies a quasi-local
Riemannian Penrose inequality, however in the context of the Einstein-Maxwell
equations, one expects that a quasi-local Riemannian Penrose inequality should
also include a contribution from the electric charge. This article builds on
ideas of Lu and Miao and of the first-named author to prove some charged
quasi-local Penrose inequalities for a class of compact manifolds with
boundary. In particular, we impose that the boundary is isometric to a closed
surface in a suitable Reissner-Nordstr\"om manifold, which serves as a
reference manifold for the quasi-local mass that we work with. In the case
where the reference manifold has zero mass and non-zero electric charge, the
lower bound on quasi-local mass is exactly the lower bound on the ADM mass
given by the charged Riemannian Penrose inequality.
