A Geometric-Analytic Proof of the Reverse Isoperimetric Inequality

We present the first geometric-analytic proof of the reverse isoperimetric inequality for black holes in any dimension. The proof holds for compact Riemannian hypersurfaces in AdS (and dS) and seems to be a generic property of black holes in the extended phase space formalism. Using Euclidean gravitational action, we show that, among all hypersurfaces of given volume, the round sphere in the $D$-dimensional (Anti-)de Sitter space maximizes the area (and hence the entropy). This analytic result is supported by a geometric argument in a $1+1+2$ decomposition of spacetime: gravitational focusing enforces a strictly negative conformal deformation, and the Sherif–Dunsby rigidity theorem then forces the deformed 3-sphere to be isometric to round 3-sphere, establishing the round sphere as the extremal surface, in fact, a maximally entropic surface. Our work establishes that the reversal of the usual isoperimetric inequality occurs due to the structure of curved background governed by Einstein's equation, underscoring the role of gravity in the reverse isoperimetric inequality for black hole horizons in (A)dS space.