  We prove the equality case of the Penrose inequality in all dimensions for
asymptotically flat hypersurfaces. It was recently proven by G. Lam that the
Penrose inequality holds for asymptotically flat graphical hypersurfaces in
Euclidean space with non-negative scalar curvature and with a minimal boundary.
Our main theorem states that if the equality holds, then the hypersurface is a
Schwarzschild solution. As part of our proof, we show that asymptotically flat
graphical hypersurfaces with a minimal boundary and non-negative scalar
curvature must be mean convex, using the argument that we developed earlier.
This enables us to obtain the ellipticity for the linearized scalar curvature
operator and to establish the strong maximum principles for the scalar
curvature equation.
