  In this paper, we generalize to higher dimensions Robinson's approach to
proving the uniqueness of connected $(3+1)$-dimensional static vacuum
asymptotically flat black hole spacetimes. In consequence, we prove geometric
inequalities for connected $(n+1)$-dimensional static vacuum asymptotically
flat black holes, recovering results by Agostiniani and Mazzieri. We also
obtain similar geometric inequalities for connected $(n+1)$-dimensional static
vacuum asymptotically flat equipotential photon surfaces and in particular
photon spheres. Assuming a natural upper bound on the total scalar curvature,
we recover the well-known uniqueness results for such black holes and
equipotential photon surfaces. We also extend the known equipotential photon
surface uniqueness results to spacetimes with negative mass. In order to
replace the role of the Cotton tensor in Robinson's proof, we introduce
techniques inspired by the analysis of Ricci solitons.
