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C N Pope - One of the best experts on this subject based on the ideXlab platform.
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black hole enthalpy and an Entropy Inequality for the thermodynamic volume
Physical Review D, 2011Co-Authors: Mirjam Cvetic, David Kubizňak, G W Gibbons, C N PopeAbstract: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 $dE= TdS + \Omega_i dJ_i + \Phi_\alpha d Q_\alpha + \Theta d \Lambda$, where $E$ is now the enthalpy of the spacetime, and $\Theta$, the thermodynamic conjugate of $\Lambda$, is proportional to an effective volume $V = -\frac{16 \pi \Theta}{D-2}$ 'inside the event horizon.' Here we calculate $\Theta$ and $V$ for a wide variety of $D$-dimensional charged rotating asymptotically 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 $R\equiv ((D-1)V/{\cal A}_{D-2})^{1/(D-1)}\, ({\cal A}_{D-2}/A)^{1/(D-2)}\ge1$, where ${\cal A}_{D-2}$ 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 $R\le 1$. Our conjectured {\it Reverse Isoperimetric Inequality} can be interpreted as the statement that the Entropy inside a horizon of a given 'volume' $V$ is maximised for Schwarzschild-AdS. The thermodynamic definition of $V$ requires a cosmological constant (or gauge coupling constant). However, except in 7 dimensions, a smooth limit exists where $\Lambda$ or $g$ goes to zero, providing a definition of $V$ even for asymptotically-flat black holes.
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black hole enthalpy and an Entropy Inequality for the thermodynamic volume
Physical Review D, 2011Co-Authors: Mirjam Cvetic, G W Gibbons, D Kubizňak, C N PopeAbstract: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 $dE= TdS + \Omega_i dJ_i + \Phi_\alpha d Q_\alpha + \Theta d \Lambda$, where $E$ is now the enthalpy of the spacetime, and $\Theta$, the thermodynamic conjugate of $\Lambda$, is proportional to an effective volume $V = -\frac{16 \pi \Theta}{D-2}$ 'inside the event horizon.' Here we calculate $\Theta$ and $V$ for a wide variety of $D$-dimensional charged rotating asymptotically 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 $R\equiv ((D-1)V/{\cal A}_{D-2})^{1/(D-1)}\, ({\cal A}_{D-2}/A)^{1/(D-2)}\ge1$, where ${\cal A}_{D-2}$ 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 $R\le 1$. Our conjectured {\it Reverse Isoperimetric Inequality} can be interpreted as the statement that the Entropy inside a horizon of a given 'volume' $V$ is maximised for Schwarzschild-AdS. The thermodynamic definition of $V$ requires a cosmological constant (or gauge coupling constant). However, except in 7 dimensions, a smooth limit exists where $\Lambda$ or $g$ goes to zero, providing a definition of $V$ even for asymptotically-flat black holes.
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black hole enthalpy and an Entropy Inequality for the thermodynamic volume
Physical Review D, 2011Co-Authors: Mirjam Cvetic, David Kubizňak, G W Gibbons, C N PopeAbstract:In a theory where the cosmological constant $\ensuremath{\Lambda}$ 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 $dE=TdS+{\ensuremath{\Omega}}_{i}d{J}_{i}+{\ensuremath{\Phi}}_{\ensuremath{\alpha}}d{Q}_{\ensuremath{\alpha}}+\ensuremath{\Theta}d\ensuremath{\Lambda}$, where $E$ is now the enthalpy of the spacetime, and $\ensuremath{\Theta}$, the thermodynamic conjugate of $\ensuremath{\Lambda}$, is proportional to an effective volume $V=\ensuremath{-}\frac{16\ensuremath{\pi}\ensuremath{\Theta}}{D\ensuremath{-}2}$ ``inside the event horizon.'' Here we calculate $\ensuremath{\Theta}$ 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 $R\ensuremath{\equiv}\phantom{\rule{0ex}{0ex}}((D\ensuremath{-}1)V/{\mathcal{A}}_{D\ensuremath{-}2}{)}^{1/(D\ensuremath{-}1)}({\mathcal{A}}_{D\ensuremath{-}2}/A{)}^{1/(D\ensuremath{-}2)}\ensuremath{\ge}1$, where ${\mathcal{A}}_{D\ensuremath{-}2}$ is the volume of the unit ($D\ensuremath{-}2$) 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 ($D\ensuremath{-}1$) space bounded by a surface of area $A$, for which $R\ensuremath{\le}1$. 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 $\ensuremath{\Lambda}$ or $g$ goes to zero, providing a definition of $V$ even for asymptotically flat black holes.
Benoit Perthame - One of the best experts on this subject based on the ideXlab platform.
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general relative Entropy Inequality an illustration on growth models
Journal de Mathématiques Pures et Appliquées, 2005Co-Authors: Stephane Mischler, Philippe Michel, Benoit PerthameAbstract:Abstract We introduce the notion of General Relative Entropy Inequality for several linear PDEs. This concept extends to equations that are not conservation laws, the notion of relative Entropy for conservative parabolic, hyperbolic or integral equations. These are particularly natural in the context of biological applications where birth and death can be described by zeroth order terms. But the concept also has applications to more general growth models as the fragmentation equations. We give several types of applications of the General Relative Entropy Inequality: a priori estimates and existence of solution, long time asymptotic to a steady state, attraction to periodic solutions for periodic forcing.
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general relative Entropy Inequality an illustration on growth models
Journal de Mathématiques Pures et Appliquées, 2005Co-Authors: Stephane Mischler, Philippe Michel, Benoit PerthameAbstract:Abstract We introduce the notion of General Relative Entropy Inequality for several linear PDEs. This concept extends to equations that are not conservation laws, the notion of relative Entropy for conservative parabolic, hyperbolic or integral equations. These are particularly natural in the context of biological applications where birth and death can be described by zeroth order terms. But the concept also has applications to more general growth models as the fragmentation equations. We give several types of applications of the General Relative Entropy Inequality: a priori estimates and existence of solution, long time asymptotic to a steady state, attraction to periodic solutions for periodic forcing.
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a consistent bgk type model for gas mixtures
Journal of Statistical Physics, 2002Co-Authors: Pierre Andries, Kazuo Aoki, Benoit PerthameAbstract:We introduce a relaxation collision operator for a mixture of gases which satisfies several fundamental properties. Different BGK type collision operators for gas mixtures have been introduced earlier but none of them could satisfy all the basic physical properties: positivity, correct exchange coefficients, Entropy Inequality, indifferentiability principle. We show that all those properties are verified for our model, and we derive its Navier–Stokes limit by a Chapman–Enskog expansion.
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the gaussian bgk model of boltzmann equation with small prandtl number
European Journal of Mechanics B-fluids, 2000Co-Authors: Pierre Andries, Patrick Le Tallec, J P Perlat, Benoit PerthameAbstract:Abstract In this paper we prove the Entropy Inequality for the Gaussian-BGK model of Boltzmann equation. This model, also called ellipsoidal statistical model, was introduced in order to fit realistic values of the transport coefficients (Prandtl number, second viscosity) in the Navier–Stokes approximation, which cannot be achieved by the usual relaxation towards isotropic Maxwellians introduced in standard BGK models. Moreover, we introduce new entropic kinetic models for polyatomic gases which suppress the internal energy variable in the phase space by using two distribution functions (one for particles mass and one for their internal energy). This reduces the cost of their numerical solution while keeping a kinetic description well adapted to desequilibrium regions.
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Kinetic Schemes for Saint-Venant Equations with Source Terms on Unstructured Grids
2000Co-Authors: Emmanuel Audusse, Marie-odile Bristeau, Benoit PerthameAbstract:We consider the Saint-Venant (or Shallow Water) system which is an usual model to describe the flows in rivers or coastal areas. This hyperbolic system of conservation laws is solved on unstructured meshes by a kinetic scheme based on a finite volume approach. An important property of this scheme is the preservation of the water height positivity when applications with dry areas are considered. Following some hypothesis an Entropy Inequality is proved. The standard kinetic scheme is modified to deal with varying bed slope and particularly to preserve equilibrium states such as still water. Moreover the source terms due to the arbitrary bottom topography have to be discretized in such a way to balance the flux gradients for these equilibriums. We illustrate the properties of the scheme on different test cases for which exact solutions are available and on more realistic applications.
David Kubizňak - One of the best experts on this subject based on the ideXlab platform.
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Entropy Inequality violations from ultraspinning black holes
Physical Review Letters, 2015Co-Authors: Robie A Hennigar, Robert B Mann, David KubizňakAbstract:We construct a new class of rotating anti-de Sitter (AdS) black hole solutions with noncompact event horizons of finite area in any dimension and study their thermodynamics. In four dimensions these black holes are solutions to gauged supergravity. We find that their Entropy exceeds the maximum implied from the conjectured reverse isoperimetric Inequality, which states that for a given thermodynamic volume, the black hole Entropy is maximized for Schwarzschild-AdS space. We use this result to suggest more stringent conditions under which this conjecture may hold.
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black hole enthalpy and an Entropy Inequality for the thermodynamic volume
Physical Review D, 2011Co-Authors: Mirjam Cvetic, David Kubizňak, G W Gibbons, C N PopeAbstract: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 $dE= TdS + \Omega_i dJ_i + \Phi_\alpha d Q_\alpha + \Theta d \Lambda$, where $E$ is now the enthalpy of the spacetime, and $\Theta$, the thermodynamic conjugate of $\Lambda$, is proportional to an effective volume $V = -\frac{16 \pi \Theta}{D-2}$ 'inside the event horizon.' Here we calculate $\Theta$ and $V$ for a wide variety of $D$-dimensional charged rotating asymptotically 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 $R\equiv ((D-1)V/{\cal A}_{D-2})^{1/(D-1)}\, ({\cal A}_{D-2}/A)^{1/(D-2)}\ge1$, where ${\cal A}_{D-2}$ 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 $R\le 1$. Our conjectured {\it Reverse Isoperimetric Inequality} can be interpreted as the statement that the Entropy inside a horizon of a given 'volume' $V$ is maximised for Schwarzschild-AdS. The thermodynamic definition of $V$ requires a cosmological constant (or gauge coupling constant). However, except in 7 dimensions, a smooth limit exists where $\Lambda$ or $g$ goes to zero, providing a definition of $V$ even for asymptotically-flat black holes.
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black hole enthalpy and an Entropy Inequality for the thermodynamic volume
Physical Review D, 2011Co-Authors: Mirjam Cvetic, David Kubizňak, G W Gibbons, C N PopeAbstract:In a theory where the cosmological constant $\ensuremath{\Lambda}$ 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 $dE=TdS+{\ensuremath{\Omega}}_{i}d{J}_{i}+{\ensuremath{\Phi}}_{\ensuremath{\alpha}}d{Q}_{\ensuremath{\alpha}}+\ensuremath{\Theta}d\ensuremath{\Lambda}$, where $E$ is now the enthalpy of the spacetime, and $\ensuremath{\Theta}$, the thermodynamic conjugate of $\ensuremath{\Lambda}$, is proportional to an effective volume $V=\ensuremath{-}\frac{16\ensuremath{\pi}\ensuremath{\Theta}}{D\ensuremath{-}2}$ ``inside the event horizon.'' Here we calculate $\ensuremath{\Theta}$ 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 $R\ensuremath{\equiv}\phantom{\rule{0ex}{0ex}}((D\ensuremath{-}1)V/{\mathcal{A}}_{D\ensuremath{-}2}{)}^{1/(D\ensuremath{-}1)}({\mathcal{A}}_{D\ensuremath{-}2}/A{)}^{1/(D\ensuremath{-}2)}\ensuremath{\ge}1$, where ${\mathcal{A}}_{D\ensuremath{-}2}$ is the volume of the unit ($D\ensuremath{-}2$) 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 ($D\ensuremath{-}1$) space bounded by a surface of area $A$, for which $R\ensuremath{\le}1$. 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 $\ensuremath{\Lambda}$ or $g$ goes to zero, providing a definition of $V$ even for asymptotically flat black holes.
Cass T Miller - One of the best experts on this subject based on the ideXlab platform.
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thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems 5 single fluid phase transport
Advances in Water Resources, 2009Co-Authors: William G Gray, Cass T MillerAbstract:This work is the fifth in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in multiscale porous medium systems. The general TCAT framework and the mathematical foundation presented in previous works are used to develop models that describe species transport and single-fluid-phase flow through a porous medium system in varying physical regimes. Classical irreversible thermodynamics formulations for species in fluids, solids, and interfaces are developed. Two different approaches are presented, one that makes use of a momentum equation for each entity along with constitutive relations for species diffusion and dispersion, and a second approach that makes use of a momentum equation for each species in an entity. The alternative models are developed by relying upon different approaches to constrain an Entropy Inequality using mass, momentum, and energy conservation equations. The resultant constrained Entropy Inequality is simplified and used to guide the development of closed models. Specific instances of dilute and non-dilute systems are examined and compared to alternative formulation approaches.
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thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems 1 motivation and overview
Advances in Water Resources, 2005Co-Authors: William G Gray, Cass T MillerAbstract:We give several examples of weaknesses in classical, empirically derived models of transport phenomena in porous medium systems. We also place recent attempts to develop improved multiscale porous medium models using averaging theory in context and note deficiencies in these approaches. These deficiencies are found to arise in part from the manner in which thermodynamics is introduced into a constrained Entropy Inequality, which is used to guide the formation of closed models. Because of this, we briefly examine several established thermodynamic approaches and outline a framework to develop macroscale models that retain consistency with microscale physics and thermodynamics. This framework will be detailed and applied in future papers in this series.
Mirjam Cvetic - One of the best experts on this subject based on the ideXlab platform.
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black hole enthalpy and an Entropy Inequality for the thermodynamic volume
Physical Review D, 2011Co-Authors: Mirjam Cvetic, David Kubizňak, G W Gibbons, C N PopeAbstract: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 $dE= TdS + \Omega_i dJ_i + \Phi_\alpha d Q_\alpha + \Theta d \Lambda$, where $E$ is now the enthalpy of the spacetime, and $\Theta$, the thermodynamic conjugate of $\Lambda$, is proportional to an effective volume $V = -\frac{16 \pi \Theta}{D-2}$ 'inside the event horizon.' Here we calculate $\Theta$ and $V$ for a wide variety of $D$-dimensional charged rotating asymptotically 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 $R\equiv ((D-1)V/{\cal A}_{D-2})^{1/(D-1)}\, ({\cal A}_{D-2}/A)^{1/(D-2)}\ge1$, where ${\cal A}_{D-2}$ 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 $R\le 1$. Our conjectured {\it Reverse Isoperimetric Inequality} can be interpreted as the statement that the Entropy inside a horizon of a given 'volume' $V$ is maximised for Schwarzschild-AdS. The thermodynamic definition of $V$ requires a cosmological constant (or gauge coupling constant). However, except in 7 dimensions, a smooth limit exists where $\Lambda$ or $g$ goes to zero, providing a definition of $V$ even for asymptotically-flat black holes.
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black hole enthalpy and an Entropy Inequality for the thermodynamic volume
Physical Review D, 2011Co-Authors: Mirjam Cvetic, G W Gibbons, D Kubizňak, C N PopeAbstract: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 $dE= TdS + \Omega_i dJ_i + \Phi_\alpha d Q_\alpha + \Theta d \Lambda$, where $E$ is now the enthalpy of the spacetime, and $\Theta$, the thermodynamic conjugate of $\Lambda$, is proportional to an effective volume $V = -\frac{16 \pi \Theta}{D-2}$ 'inside the event horizon.' Here we calculate $\Theta$ and $V$ for a wide variety of $D$-dimensional charged rotating asymptotically 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 $R\equiv ((D-1)V/{\cal A}_{D-2})^{1/(D-1)}\, ({\cal A}_{D-2}/A)^{1/(D-2)}\ge1$, where ${\cal A}_{D-2}$ 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 $R\le 1$. Our conjectured {\it Reverse Isoperimetric Inequality} can be interpreted as the statement that the Entropy inside a horizon of a given 'volume' $V$ is maximised for Schwarzschild-AdS. The thermodynamic definition of $V$ requires a cosmological constant (or gauge coupling constant). However, except in 7 dimensions, a smooth limit exists where $\Lambda$ or $g$ goes to zero, providing a definition of $V$ even for asymptotically-flat black holes.
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black hole enthalpy and an Entropy Inequality for the thermodynamic volume
Physical Review D, 2011Co-Authors: Mirjam Cvetic, David Kubizňak, G W Gibbons, C N PopeAbstract:In a theory where the cosmological constant $\ensuremath{\Lambda}$ 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 $dE=TdS+{\ensuremath{\Omega}}_{i}d{J}_{i}+{\ensuremath{\Phi}}_{\ensuremath{\alpha}}d{Q}_{\ensuremath{\alpha}}+\ensuremath{\Theta}d\ensuremath{\Lambda}$, where $E$ is now the enthalpy of the spacetime, and $\ensuremath{\Theta}$, the thermodynamic conjugate of $\ensuremath{\Lambda}$, is proportional to an effective volume $V=\ensuremath{-}\frac{16\ensuremath{\pi}\ensuremath{\Theta}}{D\ensuremath{-}2}$ ``inside the event horizon.'' Here we calculate $\ensuremath{\Theta}$ 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 $R\ensuremath{\equiv}\phantom{\rule{0ex}{0ex}}((D\ensuremath{-}1)V/{\mathcal{A}}_{D\ensuremath{-}2}{)}^{1/(D\ensuremath{-}1)}({\mathcal{A}}_{D\ensuremath{-}2}/A{)}^{1/(D\ensuremath{-}2)}\ensuremath{\ge}1$, where ${\mathcal{A}}_{D\ensuremath{-}2}$ is the volume of the unit ($D\ensuremath{-}2$) 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 ($D\ensuremath{-}1$) space bounded by a surface of area $A$, for which $R\ensuremath{\le}1$. 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 $\ensuremath{\Lambda}$ or $g$ goes to zero, providing a definition of $V$ even for asymptotically flat black holes.