A proof of the reverse isoperimetric inequality using a geometric-analytic approach

We present a proof of the reverse isoperimetric inequality - a central conjecture in extended black hole thermodynamics - for black holes in Einstein gravity with $D\ge 4$, employing a two-pronged geometric-analytic method. Our analysis shows that the reversal of the usual isoperimetric inequality originates from the structure of curved backgrounds governed by Einstein’s equations, thereby underscoring the fundamental role of gravity in the reverse isoperimetric property of AdS black hole horizons.