Reverse Isoperimetric Conjecture

• The reverse isoperimetric conjecture is a framework of inverse extremal problems where one minimizes volume or maximizes boundary measures under fixed auxiliary constraints.
• Recent developments extend the theory to dual Brunn–Minkowski formulations, curvature-constrained bodies, and even black-hole thermodynamics, broadening its interdisciplinary impact.
• Methodologies such as affine normalization, curvature bounds, and variational techniques reveal non-classical optimizers like cubes, lenses, and sausage bodies in both deterministic and stochastic settings.

Searching arXiv for recent and foundational papers on the reverse isoperimetric conjecture and related inequalities. First search: general reverse isoperimetric papers. The reverse isoperimetric conjecture denotes a family of reverse extremal problems in which the classical isoperimetric direction is inverted: instead of maximizing volume or area for fixed boundary measure, one seeks maximizers of perimeter or surface area, or minimizers of volume or area, within a constrained class. In convex geometry the prototype is Ball’s reverse isoperimetric inequality, while recent work has developed relative, dual, stochastic, curvature-constrained, Gaussian, and thermodynamic variants. In black-hole thermodynamics, the standard formulation asserts that at fixed thermodynamic volume, Schwarzschild–AdS black holes maximize entropy (Sadovsky et al., 29 May 2025, Kumar, 25 Jun 2026).

1. Principal formulations

The phrase is used for several non-equivalent, but structurally related, problems. What unifies them is the presence of an auxiliary constraint—affine normalization, curvature bounds, inclusion constraints, or a thermodynamic equation of state—without which the reverse problem is typically trivial or ill-posed.

Setting	Constraint	Extremal object
Origin-symmetric convex bodies in John’s position	Fixed affine normalization	Cube $B_\infty^n$ maximizes normalized dual quermassintegrals (Sadovsky et al., 29 May 2025)
$\lambda$-concave bodies	Fixed surface area	$\lambda$-sausage body minimizes volume (Chernov et al., 2018)
$\lambda$-convex bodies	Fixed surface area	$\lambda$-convex lens minimizes volume (Drach et al., 2023, Drach et al., 9 Mar 2026)
Relative outer parallel bodies $K=M+E$	Minkowski-relative structure	Equality iff $\dim(M)\le 1$ in the reverse quermassintegral inequality (Gómez et al., 2019)
Convex sets in Gaussian plane	Fixed convex admissible class	Smooth maximizers, if they exist, have locally flat boundary (Brock et al., 27 Mar 2025)
AdS black holes	Fixed thermodynamic volume	Schwarzschild–AdS maximizes entropy (Kumar, 25 Jun 2026)

A recurrent feature is that extremizers are rarely round in the Euclidean sense. Depending on the setting, they are cubes, simplices, lenses, sausage bodies, inscribed polygons, regular hyperbolic polygons, or degenerate limits.

2. Affine and dual Brunn–Minkowski theory

A major recent development is the dual Brunn–Minkowski formulation of reverse isoperimetry. For an origin-symmetric convex body $K\subset\mathbb R^n$ with radial function $\rho_K$, the $q$-th dual quermassintegral is

$\lambda$0

and the normalized quantity is

$\lambda$1

For $\lambda$2, the dual Brunn–Minkowski inequality

$\lambda$3

was proved for origin-symmetric convex bodies, resolving Lutwak’s conjecture in this range (Sadovsky et al., 29 May 2025).

The reverse isoperimetric statement in this framework concerns bodies in John’s position. If $\lambda$4 is an origin-symmetric convex body in John’s position, then for every $\lambda$5,

$\lambda$6

equivalently $\lambda$7, with equality if and only if $\lambda$8 is a rotation of the cube $\lambda$9 (Sadovsky et al., 29 May 2025). The case $\lambda$0 recovers Ball’s volume ratio inequality, so the theorem extends the reverse isoperimetric principle from volume to the full family of dual quermassintegrals.

A complementary asymptotic direction concerns convex polytopes. Writing

$\lambda$1

the smallest possible isoperimetric quotient of an $\lambda$2-dimensional convex polytope with $\lambda$3 facets is bounded from above and from below by positive universal constant multiples of

$\lambda$4

and for origin-symmetric $\lambda$5-polytopes with $\lambda$6 vertices there is an affine image whose isoperimetric quotient is at most a universal constant multiple of

$\lambda$7

with sharpness up to universal constants (Ball et al., 17 Sep 2025). The same paper proves the weak isomorphic reverse isoperimetry conjecture for $\lambda$8-dimensional convex polytopes with $\lambda$9 facets by finding a volume-preserving affine image containing a large subbody with isoperimetric quotient at most a universal constant multiple of $\lambda$0 (Ball et al., 17 Sep 2025).

These results place reverse isoperimetry squarely within affine convexity: the extremal quantity is not merely geometric volume versus surface area, but an affine-normalized isoperimetric functional.

3. Curvature-constrained bodies

A second major branch imposes pointwise curvature constraints. For $\lambda$1-concave bodies in $\lambda$2, each boundary point supports a tangent ball of radius $\lambda$3 lying locally inside the body. Such bodies admit the decomposition

$\lambda$4

and $\lambda$5 is a $\lambda$6-sausage body if and only if $\lambda$7 (Chernov et al., 2018). In this class one has the reverse quermassintegral inequality

$\lambda$8

for every $\lambda$9, with equality if and only if $\lambda$0 is a $\lambda$1-sausage body (Chernov et al., 2018). The corresponding reverse isoperimetric corollary is

$\lambda$2

again with equality if and only if $\lambda$3 is a $\lambda$4-sausage body (Chernov et al., 2018).

For $\lambda$5-convex bodies, where the principal curvatures satisfy $\lambda$6 in the smooth case, the extremizer is instead the $\lambda$7-convex lens, the intersection of two supporting $\lambda$8-regions. In $\lambda$9, if $K=M+E$0 is $K=M+E$1-convex and $K=M+E$2 is the $K=M+E$3-convex lens with $K=M+E$4, then

$K=M+E$5

with equality if and only if $K=M+E$6 is a $K=M+E$7-convex lens (Drach et al., 2023). This confirms Borisenko’s conjecture in $K=M+E$8. The same lens-minimization statement was then proved in all three-dimensional nonflat space forms $K=M+E$9, $\dim(M)\le 1$0, again with uniqueness of the lens (Drach et al., 9 Mar 2026).

Related work sharpens the structure of possible maximizers in the dual problem of maximizing perimeter at fixed volume under $\dim(M)\le 1$1-convexity. In Euclidean, spherical, and hyperbolic space, a nontrivial compact $\dim(M)\le 1$2-convex body with largest surface area among all $\dim(M)\le 1$3-convex bodies of the same volume cannot be of class $\dim(M)\le 1$4; moreover, on the set where a maximizer is $\dim(M)\le 1$5, its smallest principal curvature is constantly $\dim(M)\le 1$6 (Hamdy et al., 4 Nov 2025). This rules out smooth maximizers and forces curvature saturation on any smooth piece.

In the planar two-sided curvature-bounded problem, $\dim(M)\le 1$7-convex bodies satisfy

$\dim(M)\le 1$8

almost everywhere, where $\dim(M)\le 1$9 is the support function and $K\subset\mathbb R^n$0 is the radius of curvature (Croce et al., 2021). Among such bodies of fixed perimeter, the minimizer of area is an $K\subset\mathbb R^n$1-egg, whose boundary consists of two arcs of circles of radius $K\subset\mathbb R^n$2 joined by two arcs of circles of radius $K\subset\mathbb R^n$3 (Croce et al., 2021).

4. Relative, functional, and stochastic extensions

Reverse isoperimetric behavior also appears in Minkowski-relative geometry. For convex bodies $K\subset\mathbb R^n$4, the relative outer parallel body is $K\subset\mathbb R^n$5, and the relative Steiner formula is

$K\subset\mathbb R^n$6

If $K\subset\mathbb R^n$7 with $K\subset\mathbb R^n$8, then for every $K\subset\mathbb R^n$9,

$\rho_K$0

with equality if and only if $\rho_K$1 (Gómez et al., 2019). The case $\rho_K$2 yields

$\rho_K$3

where $\rho_K$4 is the relative Minkowski content (Gómez et al., 2019). The same paper derives these inequalities from the convexity of the sequence $\rho_K$5 for such Minkowski sums.

A functional higher-order extension is provided for log-concave functions. The Rogers–Shephard and Zhang projection inequalities are interpreted as reverse affine isoperimetric inequalities, and higher-order analogues are established in the log-concave setting, with the sharp combinatorial constant $\rho_K$6 and simplex extremizers (Langharst et al., 2024). Equality is characterized by simplex-type exponentials, or by characteristic functions of $\rho_K$7-dimensional simplices in the indicator-function regime (Langharst et al., 2024).

A stochastic strengthening exists in the plane. For random polytopes generated by independent uniform points in a convex body, the functionals measuring negative moments of volumes of polars are convex along RS-movements. This yields stochastic analogues of Mahler’s theorem and of the reverse Lutwak–Zhang inequality: in the centrally symmetric planar case the extremizers are parallelograms or squares, while in the nonsymmetric case the extremizer is a triangle (Bueno, 2021). The deterministic inequalities are recovered as large-$\rho_K$8 limits.

5. Planar maximization under inclusion and Gaussian constraints

In the unit disk, the reverse isoperimetric problem with inclusion constraint is completely solved. For fixed area $\rho_K$9,

$q$0

has a maximizer that is a polygon inscribed in $q$1 (Bogosel, 2024). A no free vertex theorem states that every vertex of a maximizing polygon lies on $q$2, and the optimal polygon has $q$3 sides uniquely determined by

$q$4

with central angles satisfying

$q$5

so all but one side are equal (Bogosel, 2024). The smallest angle $q$6 is determined by

$q$7

This yields a complete characterization in the disk, while for a general convex container only existence and numerical evidence are established (Bogosel, 2024).

A distinct planar reverse problem arises in Gauss space. For convex sets in $q$8, the objective is to maximize Gaussian perimeter

$q$9

among convex sets (Brock et al., 27 Mar 2025). The sharp structure theorem is local: if a perimeter maximizer is $\lambda$00 on a boundary arc, then that arc must be a straight segment (Brock et al., 27 Mar 2025). Thus smooth maximizers cannot have strictly curved smooth pieces. The paper also proves $\lambda$01 for a large class of $\lambda$02-monotone convex sets and for convex sets symmetric with respect to both coordinate axes, and shows that for convex quadrilaterals with vertices on the coordinate axes the maximizing sequence degenerates into the $\lambda$03-axis traversed twice (Brock et al., 27 Mar 2025).

These planar results show that reverse extremizers need not resemble classical smooth isoperimetric shapes. In the disk the optimizer is polygonal and rigidly parameterized by the area fraction; in the Gaussian plane, extremizing sequences flatten and may become genuinely degenerate.

6. Nonpositively curved surfaces and hyperbolic tilings

On nonpositively curved surfaces, reverse isoperimetry takes the form of lower bounds on area. For a disk $\lambda$04 with boundary length $\lambda$05 and total negative curvature

$\lambda$06

one has the sharp inequality

$\lambda$07

with equality for the corresponding disk centered at the vertex of a Euclidean cone (Katz et al., 2021). The same paper proves analogous sharp lower bounds for geodesic triangles: among triangles with the same side lengths and angles in curvature bounded above by $\lambda$08, the area-minimizer is the comparison triangle in a cone $\lambda$09 (Katz et al., 2021).

A tiling-theoretic hyperbolic analogue concerns regular $\lambda$10-gons with $\lambda$11 angles. For real $\lambda$12, let

$\lambda$13

Then a curvilinear polygonal tiling of a closed hyperbolic surface with average tile area $\lambda$14 and perimeter at most $\lambda$15 forces $\lambda$16 to be an integer and every tile to be equivalent to the regular $\lambda$17-gon $\lambda$18 (Hirsch et al., 2019). In particular, Cox’s conjecture is proved: the regular $\lambda$19-gonal tile with $\lambda$20 angles is the least-perimeter tile for its area (Hirsch et al., 2019).

These results are not phrased as Ball-type affine inequalities, but they belong to the same reverse-isoperimetric program: the extremal object is a cone model or a regular hyperbolic polygon, and the inequality gives a lower, rather than upper, bound on area for prescribed boundary data.

7. Black-hole thermodynamics and the AdS formulation

In extended black-hole thermodynamics, the cosmological constant is treated as pressure,

$\lambda$21

and the reverse isoperimetric conjecture is usually stated as: at fixed thermodynamic volume, Schwarzschild–AdS black holes maximize entropy (Kumar, 25 Jun 2026). In Einstein gravity this is written through the ratio

$\lambda$22

equivalently as an area–volume inequality (Kumar, 25 Jun 2026).

For static AdS black holes in Einstein gravity, a near-horizon identity links thermodynamic volume, Euclidean bounded volume, and butterfly velocity: $\lambda$23 Using Einstein’s equations and the null-energy condition, one obtains $\lambda$24, hence $\lambda$25, so the reverse isoperimetric inequality follows for static AdS black holes and yields an upper bound on butterfly velocity (Feng et al., 2017).

Recent work reformulates the conjecture as a stability theorem. The bulk Hollands–Wald canonical energy vanishes along exact stationary families, so it is not the full entropy Hessian. The missing curvature is supplied by a constrained asymptotic charge Hessian, leading to the boundary-completed canonical energy

$\lambda$26

and the fixed-volume Hessian identity

$\lambda$27

Positivity of $\lambda$28 implies entropy concavity on admissible fixed-volume components, hence

$\lambda$29

with equality controlled by a rigidity sector $\lambda$30 (Kumar, 25 Jun 2026). In this interpretation, known violations are reclassified as failures of compactness, positivity, or rigidity rather than failures of the variational mechanism itself (Kumar, 25 Jun 2026).

The conjecture is not universal once one leaves Einstein gravity. In Einstein–Horndeski–Maxwell gravity with Horndeski axions, charged AdS planar black holes can violate

$\lambda$31

for negative Horndeski coupling $\lambda$32, even when the no-ghost condition $\lambda$33 holds (Feng et al., 2017). By contrast, a 2025 geometric-analytic proof claims that for compact Riemannian hypersurfaces in AdS, the round sphere maximizes area at fixed volume, using Euclidean gravitational action together with a rigidity argument based on gravitational focusing and a conformal deformation in a $\lambda$34 decomposition (Kumar, 18 Aug 2025).

Across its geometric and thermodynamic incarnations, the reverse isoperimetric conjecture is therefore best understood not as a single theorem, but as a broad extremal principle. Its sharp formulations depend crucially on the admissible class, and its equality cases reveal a consistent pattern: extremizers are governed by active constraints, often becoming cubes, simplices, lenses, sausage bodies, piecewise-circular polygons, flat Gaussian boundaries, or Schwarzschild–AdS reference states rather than unconstrained smooth round bodies.