Black hole enthalpy and an entropy inequality for the thermodynamic volume

In a theory where the cosmological constant Lambda or the gauge coupling constant g arises as the vacuum expectation value, its variation should be included in the first law of thermodynamics for black holes. This becomes dE = TdS + Omega(i)dJ(i) + Phi(alpha)dQ(alpha) + Theta d Lambda, where E is now the enthalpy of the spacetime, and Theta, the thermodynamic conjugate of Lambda, is proportional to an effective volume V = -16 pi Theta/D-2 "inside the event horizon." Here we calculate Theta and V for a wide variety of D-dimensional charged rotating asymptotically anti-de Sitter (AdS) black hole spacetimes, using the first law or the Smarr relation. We compare our expressions with those obtained by implementing a suggestion of Kastor, Ray, and Traschen, involving Komar integrals and Killing potentials, which we construct from conformal Killing-Yano tensors. We conjecture that the volume V and the horizon area A satisfy the inequality R equivalent to ((D - 1)V/A(D-2))(1/(D-1))(A(D-2)/A)(1/(D-2)) >= 1, whereA(D-2) is the volume of the unit (D - 2) sphere, and we show that this is obeyed for a wide variety of black holes, and saturated for Schwarzschild-AdS. Intriguingly, this inequality is the "inverse" of the isoperimetric inequality for a volume V in Euclidean (D - 1) space bounded by a surface of area A, for which R <= 1. Our conjectured reverse isoperimetric inequality can be interpreted as the statement that the entropy inside a horizon of a given "volume" V is maximized for Schwarzschild-AdS. The thermodynamic definition of V requires a cosmological constant (or gauge coupling constant). However, except in seven dimensions, a smooth limit exists where Lambda or g goes to zero, providing a definition of V even for asymptotically flat black holes.