On the Classical, Penrose, and Reverse Isoperimetric Inequalities in Black Holes: Insights From AdS to Flat Riemannian Backgrounds
Abstract
A collection of evidence is presented showing that the conjectured reverse isoperimetric inequality (RII) for asymptotically anti-de Sitter (AdS) as well as de Sitter (dS) black holes, cal RRII ≥ 1, is of fundamental importance and its violation (cal RRII < 1) gives rise to either thermodynamic instabilities, or physically unreasonable solutions, or naked singularities. We show that AdS black holes violating reverse isoperimetric inequality, known as superentropic black holes, satisfy neither mechanical stability requirement of κT geqslant κS geqslant 0 nor thermal stability requirement of CP geqslant CV geqslant 0. This property makes them thermodynamically unstable for the whole range of parameter space and explains all the diverse behaviors so far reported. We conjecture this statement holds for all superentropic black holes. We confirm that there is no counterexample to the thermodynamic instability conjecture of superentropic black holes and extend this to dS space by presenting the first example of superentropicity in dS space. By bringing evidence, we then propose two other conjectures: 1) there is no asymptotically flat limit of superentropic black holes, and 2) there exist two stronger versions of Penrose isoperimetric inequality (PII) for asymptotically flat black holes; the first is the asymptotically flat limit of the RII and the second is a new inequality in terms of the ADM mass and the thermodynamics volume, that we call thermo-volumetric inequality, obtained via insights from the Λ→ 0 limit of extended black hole thermodynamics in AdS. We show that the Penrose isoperimetric inequality (PII) is weaker than the reverse isoperimetric inequality (RII), as demonstrated explicitly for D=4,5 and for all black hole families we have examined.
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BibTeX
@misc{mann2026classical,
title = {{On the Classical, Penrose, and Reverse Isoperimetric Inequalities in Black Holes: Insights From AdS to Flat Riemannian Backgrounds}},
author = {Robert B. Mann and Behnam Pourhassan and Ali Dehghani},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14108},
eprint = {2609.14108},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14108},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Classical, Penrose, and Reverse Isoperimetric Inequalities in Black Holes: Insights From AdS to Flat Riemannian Backgrounds
AU - Robert B. Mann
AU - Behnam Pourhassan
AU - Ali Dehghani
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.14108
UR - https://arxiv.org/abs/2609.14108
AB - A collection of evidence is presented showing that the conjectured reverse isoperimetric inequality (RII) for asymptotically anti-de Sitter (AdS) as well as de Sitter (dS) black holes, cal R_RII ≥ 1, is of fundamental importance and its violation (cal R_RII < 1) gives rise to either thermodynamic instabilities, or physically unreasonable solutions, or naked singularities. We show that AdS black holes violating reverse isoperimetric inequality, known as superentropic black holes, satisfy neither mechanical stability requirement of κ_T geqslant κ_S geqslant 0 nor thermal stability requirement of C_P geqslant C_V geqslant 0. This property makes them thermodynamically unstable for the whole range of parameter space and explains all the diverse behaviors so far reported. We conjecture this statement holds for all superentropic black holes. We confirm that there is no counterexample to the thermodynamic instability conjecture of superentropic black holes and extend this to dS space by presenting the first example of superentropicity in dS space. By bringing evidence, we then propose two other conjectures: 1) there is no asymptotically flat limit of superentropic black holes, and 2) there exist two stronger versions of Penrose isoperimetric inequality (PII) for asymptotically flat black holes; the first is the asymptotically flat limit of the RII and the second is a new inequality in terms of the ADM mass and the thermodynamics volume, that we call thermo-volumetric inequality, obtained via insights from the Λ→ 0 limit of extended black hole thermodynamics in AdS. We show that the Penrose isoperimetric inequality (PII) is weaker than the reverse isoperimetric inequality (RII), as demonstrated explicitly for D=4,5 and for all black hole families we have examined.
ER -