Revised Reverse Isoperimetric Inequality

• The revised reverse isoperimetric inequality is a refined principle that bounds black hole entropy and rectifies earlier violations seen in ultraspinning cases.
• It imposes strict parameter constraints on ultraspinning and charged black holes, ensuring that Kerr–AdS solutions yield the maximal entropy at fixed charges and volume.
• The RRII underpins consistent thermodynamic laws in extended phase spaces and guides the accurate construction of physically admissible black holes.

A revised reverse isoperimetric inequality (RRII) addresses fundamental bounds on the entropy of black holes—especially in the presence of ultraspinning or "super-entropic" black holes, whose horizon geometries and thermodynamics deviate sharply from classic cases. The RRII emerges as a response to violations of the original reverse isoperimetric inequality (RII) observed in recent constructions, and imposes refined constraints even in these exotic regimes.

1. Foundations and Motivation

The original reverse isoperimetric inequality was formulated to encode the expectation that, at fixed thermodynamic volume, the entropy (area) of a black hole in AdS should not exceed that of the Schwarzschild–AdS solution, with the conjectured bound

$$\mathcal R \equiv \left[\frac{(d-1)V}{\omega_{d-2}}\right]^{1/(d-1)}\left[\frac{\omega_{d-2}}{A}\right]^{1/(d-2)} \ge 1$$

where $V$ is the thermodynamic volume, $A$ is the horizon area, and $\omega_{d-2}$ is the volume of the unit $(d-2)$-sphere. This inequality is saturated by static Schwarzschild–AdS black holes (Hennigar et al., 2015, Xu, 2020).

Recent discovery of ultraspinning (super-entropic) black holes—whose optimal angular velocities reach the AdS speed of light and which possess finite-area but noncompact (e.g., Spheres with punctures) horizons—showed that these objects can have $\mathcal R < 1$, violating the conjectured bound (Hennigar et al., 2015, Di, 30 Jan 2026). Such violations highlight the need to revise or nuance the isoperimetric principle for general classes of black holes.

2. Formulation of the Revised Reverse Isoperimetric Inequality

The revised reverse isoperimetric inequality (RRII) states:

Among all rotating AdS black holes of fixed total mass $M$, angular momenta $J_i$, and thermodynamic volume $V$, the maximally symmetric Kerr–AdS solution has the largest entropy (area): $> A(M, J_i, V) \leq A_{\text{Kerr}}(M, J_i, V) >$ Super-entropic black holes always satisfy this refined bound, even if they violate the original RII (Di, 30 Jan 2026).

This version replaces "volume at fixed area" by a variational statement at fixed conserved charges and fixes the reference solution to be Kerr–AdS in the appropriate class of black holes. In this sense, any deviation from the regular Kerr–AdS geometry—such as an ultraspinning limit, inclusion of additional charges, or noncompact horizon—exhibits entropy strictly below the maximal Kerr–AdS solution at each point in the allowed parameter space.

3. Parameter Constraints Imposed by the RRII

The formal application of the RRII to explicit black hole families, most notably in four-dimensional super-entropic Kerr–AdS or charged analogs (Kerr–Newman–AdS, Dyonic Kerr–Sen–AdS), yields a hierarchy of constraints:

• Upper bound on the ultraspinning parameter $\mu$: For fixed mass, charge, and AdS scale, there exists $\mu \leq \mu_{\rm max}(m, q, l)$.
• Strengthened lower bound on mass $m$: Beyond basic requirements for horizon existence, the RRII imposes a minimal mass for given charge and AdS radius.
• Upper bounds on charge $q$ and AdS radius $l$: Given the other parameters, only bounded values of $q$, $l$ yield acceptable black holes.
• Exclusion of certain parameter regions: Not all combinations of $(m, q, l, \mu)$ correspond to physical black holes; the RRII restricts the allowed domain (Di, 30 Jan 2026).

Hence, while super-entropic black holes routinely violate the original RII ($\mathcal R < 1$), the RRII ensures that their entropy is always submaximal for fixed extensive charges.

4. Thermodynamics and the RRII: Consistency and Novel Features

The RRII has significant implications for black hole thermodynamics, especially in the context of consistent definitions of the first law and Smarr relations in the ultraspinning regime. For ultraspinning Kerr–AdS (and relatives), the Iyer–Wald formalism confirms that the first law in the extended phase space (with variable cosmological constant) remains valid, provided an additional chemical potential conjugate to the periodicity $\mu$ of the compactified, noncompact direction is included (Di, 30 Jan 2026, Hennigar et al., 2015). The entropy/volume structure, along with the RRII, precludes unphysical behavior and guides the construction of physically admissible black holes.

The table below summarizes the parameter bounds from the RRII in the ultraspinning Kerr–AdS sector (Di, 30 Jan 2026):

RRII-Imposed Bound	Physical Interpretation	Source
$\mu \leq \mu_{\rm max}(m, q, l)$	Max. compactification period for physical solution	(Di, 30 Jan 2026)
$m \geq m_{\rm crit}(q, l)$	Minimal mass beyond horizon formation	(Di, 30 Jan 2026)
$q \leq q_{\rm max}(m, l)$	Maximal charge allowed for a given mass/radius	(Di, 30 Jan 2026)
$l \leq l_{\rm max}(m, q)$	Maximal AdS scale for fixed mass/charge	(Di, 30 Jan 2026)

5. Dimensional, Charge, and Ensemble Dependence of the RRII

The detailed phenomenology of the RRII is sensitive to spacetime dimension $d$ and to inclusion of gauge charges. For $d>5$, new features such as critical points and intermediate stability bands arise in heat capacities and phase structure, while electric charge tends to reduce or split thermodynamically stable regions. The RRII remains valid throughout, but the shapes of allowed domains and the precise maxima of entropy are deformed as $d$ and $q$ vary (Di, 30 Jan 2026).

Additionally, the existence of multiple solution branches—distinct by values of the compactification parameter $\mu$ or by different geometrical/topological sectors—means that the RRII enforces cross-branch bounds, further refining the phase structure.

6. RRII in Charged, Gauged, and Noncompact Black Hole Families

Charged and/or gauged black holes display nontrivial interplay between the RRII and the possible violation of the original RII:

• Ultraspinning dyonic Kerr–Sen–AdS$_4$: The isoperimetric ratio $\mathcal R$ may be less than, equal to, or greater than unity, depending sensitively on the (dilaton, axion) charge, AdS radius, and horizon radius. Only below a threshold in total charge does the solution become super-entropic ($\mathcal R<1$), but the RRII is always obeyed (Sakti et al., 2022, Wu et al., 2020).
• Ultraspinning Chow black holes in six-dimensional gauged supergravity: For singly-rotating solutions, sub-entropic ($\mathcal R\geq1$) and super-entropic ($\mathcal R<1$) branches coexist depending on the interplay between mass, charges, and rotation, while purely ultraspinning Kerr–AdS black holes are always super-entropic (Wu et al., 2021).

These results demonstrate that the RRII supersedes the original RII as the correct entropy bound: regardless of how severe the violation of the old bound, the generalized Kerr–AdS solution remains the entropy maximizer at fixed charges and volume—a principle compatible even with highly non-trivial topologies or non-compact horizons.

7. Physical Significance and Open Problems

The confirmed validity of the revised reverse isoperimetric inequality in ultraspinning and super-entropic sectors establishes a robust maximal-entropy principle for AdS black holes under broad conditions. This bound survives the exotic geometric and thermodynamic structures that emerge in these regimes: non-compact horizons, topologies with punctures, and multifaceted ensemble structure in extended thermodynamics (Di, 30 Jan 2026).

However, the RRII remains a conjecture for the maximally general class of black holes. Its precise range of validity—across gauged, charged, multi-rotational, or higher-derivative extensions—remains a subject of ongoing research, as does the deeper geometrical or microphysical reason underlying its universality. A rigorous proof is still lacking, particularly outside the field of solutions that can be linked to the Kerr–AdS class via ultraspinning or related limits.

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References:

(Hennigar et al., 2015, Di, 30 Jan 2026, Sakti et al., 2022, Wu et al., 2020, Wu et al., 2021, Xu, 2020)