Penrose's inequality for asymptotically Euclidean manifolds

Let $(M^d,g)$ be a complete asymptotically Euclidean manifold of real dimension $d$ with non-negative scalar curvature and an outermost minimal hypersurface $\Sigma$. Write $m_{ADM}$ for its ADM mass, $V(\Sigma)$ for the volume of $\Sigma$, and

$$V_E(\Sigma_1)=\frac{2\pi^d}{(d-1)!}$$

for the volume of the Euclidean unit hypersphere. Penrose's inequality. The ADM mass satisfies

$$m_{ADM}\geq \frac{1}{2}\left(\frac{V(\Sigma)}{V_E(\Sigma_1)}\right)^{\frac{d-2}{d-1}}.$$

Moreover, equality holds if and only if $M^d$ is isometric to a spatial Schwarzschild manifold outside its horizon. This is the Riemannian Penrose inequality, proved up to dimension eight in the cited work; the source presents it as a classical conjecture in the general Riemannian setting.