REVIEW 3 major objections 3 minor 10 references
Two properties of optimisers for the reverse isoperimetric problem
T0 review · 3 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash
Pith's one-line read The reverse isoperimetric problem has no C^2 maximizers: any smooth patch of a maximizer must have its smallest principal curvature equal to λ.
desk verdict Genuinely new variational result with a real gap in the hyperbolic λ≤1 case; the Euclidean and spherical parts look solid. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The key object is the stability operator T = Δ + Ric(ν,ν) + |A|^2 acting on normal variations of a constant-mean-curvature patch, together with the sufficient condition for strong stability: existence of a positive function u with T(u) ≤ 0. In the space forms this condition is verified by exhibiting a Killing field X with ⟨ν, X⟩ > 0. Proposition 3.5 is the load-bearing variational step: it builds two-parameter deformations supported in a strictly λ-convex patch, uses the implicit function theorem to fix volume, and shows the first variation of area is positive if mean curvature varies, or the second variation is positive (via −∫ vT(v) > 0) if mean curvature is constant. This is what forces t
What would settle it
Exhibit a smooth (C^2) λ-convex body in hyperbolic space with λ < 1 whose boundary has a horosphere patch and compute the second variation of area for all volume-preserving compact variations: if any such variation has non-positive second variation while the first vanishes, the improvement lemma fails for that patch and the no-smooth-maximizer claim for λ < 1 collapses. More directly, producing any smooth λ-convex body whose area is greater than every λ-convex body of the same volume would refute the theorem.
Extended reading notes
Core claim
The paper's central claim is Theorem 1.2: for any λ > 0 and any nontrivial compact λ-convex body K in Euclidean, spherical or hyperbolic space that maximizes surface area among all λ-convex bodies of the same volume, the boundary ∂K is not C^2, and—when λ lies in the parameter range for which a geodesic sphere of curvature λ exists—the smallest principal curvature on every C^2 patch of ∂K is constantly λ. The proof proceeds by contradiction: if a C^2 patch were strictly λ-convex (smallest curvature > λ), Proposition 3.5 constructs a compactly supported, volume-preserving deformation that strictly increases area, so K could not be a maximizer. A geometric lemma first shows that every nontrivi
Load-bearing premise
The proof's local improvement step needs every constant-mean-curvature smooth patch of a maximizer's boundary to be strictly stable under volume-preserving perturbations; in hyperbolic space this is established only for patches that are not horospheres, so for λ ≤ 1 the claim outruns the stability argument as written.
Editorial extensions
If this is right
- If the theorem is correct, the reverse isoperimetric problem admits no C^2 maximizer in Euclidean, spherical, or hyperbolic space for any λ > 0; maximizers, if they exist, must be singular.
- Any smooth portion of a maximizer's boundary is curvature-saturated: its smallest principal curvature is identically λ, so the boundary cannot be strictly convex at a smooth point.
- The variational obstruction also applies to spherical and hyperbolic settings, where no explicit maximizers were previously known, so the result substantially narrows the search.
- The proof shows that a volume-preserving deformation increasing area exists whenever a strictly λ-convex smooth patch is present, making such patches structurally unstable as maximizers.
Reading between the lines
- The saturation principle suggests a free-boundary picture: maximizers should be assembled from pieces of λ-umbilic hypersurfaces (geodesic spheres of curvature λ) meeting along singular creases, analogous to the known Euclidean λ-lens examples.
- One could test the mechanism numerically: running the volume-preserving area-increasing deformation on a smooth strictly λ-convex body should drive it toward a singular, curvature-saturated shape; this would offer a constructive route to maximizers in dimensions where none are explicit.
- A direct stability check for horosphere caps in hyperbolic space would determine whether the no-smoothness result for λ<1 follows from the same variational mechanism or needs a separate argument.
- The two-parameter variation technique may carry over to other shape optimization problems with non-open admissible classes, such as maximizing area under a prescribed lower bound on mean width.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the reverse isoperimetric problem in simply connected space forms: for a compact λ-convex body K of fixed volume maximizing perimeter, it claims (i) ∂K cannot be C², and (ii) on C² portions of a maximizer, the smallest principal curvature is identically λ (for λ in the range where geodesic spheres of curvature λ exist). The proof uses a rolling-ball theorem (Theorem 2.6) to show that every nontrivial λ-convex body lies in a nontrivial lens (Lemma 3.1), which is contained in a strictly smaller ball (Lemma 3.2); comparison then yields a strictly λ-convex C² patch (Corollary 3.3). The authors then construct a volume-preserving variation supported in such a patch (Proposition 3.5) which increases area, using the first variation when mean curvature is nonconstant and the second variation plus strong stability when it is constant. They conclude no C² maximizer can exist and the smallest curvature on C² pieces must be λ.
Significance. If correct, the result advances the reverse isoperimetric problem in dimension ≥3, where explicit maximizers are not generally known. The two properties — non-existence of smooth maximizers and saturation of the smallest principal curvature — are natural and elegant. The proof strategy is transparent and builds on established rolling and stability results; no parameters are fitted. The Euclidean and spherical parts are convincing. The hyperbolic λ≤1 case, however, is not fully justified as written; because the gap is localized and likely repairable, the core contribution remains valuable.
major comments (3)
- [Theorem 1.2 / Proposition 3.5] Theorem 1.2(i) states a result for all λ>0 in H^{n+1}, but the proof invokes Proposition 3.5, which is explicitly stated only for λ∈I_Σ (in the hyperbolic case λ>1). For 0<λ≤1, Corollary 3.3 provides a strictly λ-convex C² patch (the touching-sphere argument), but Proposition 3.5 cannot be applied in its stated form. No alternative variation argument is supplied. This leaves the main claim for hyperbolic λ≤1 unproved.
- [Proposition 3.4 / Proposition 3.5, Case 2] Even if Proposition 3.5 is extended to λ≤1, Case 2 needs the strict inequality -∫_Ω vT(v)>0 for compactly supported zero-mean v on the constant-mean-curvature patch. The only stability criterion provided, Proposition 3.4, explicitly excludes hyperbolic CMC hypersurfaces contained in a horosphere. A C² λ-convex body with λ<1 may have a boundary portion that is a genuine horosphere patch (all principal curvatures 1>λ); Corollary 3.3 does not rule this out. Since no stability computation for horosphere patches appears, the proof of the hyperbolic case is incomplete. (A repair is straightforward: on a horosphere T=Δ, so -∫vT(v)=∫|∇v|²>0, or one can use the dilation Killing field y∂_y in the upper half-space model to extend Proposition 3.4.)
- [Proposition 3.5, first paragraph] The admissibility of the constructed variation is asserted in one sentence ('as can be checked locally from Definition 1.1 using a case distinction'). This is a load-bearing point: the whole argument rests on K_{t,s} remaining λ-convex. Please give the detailed proof that a sufficiently small C² deformation supported in a strictly λ-convex patch preserves the λ-convexity condition, including at the boundary of the support.
minor comments (3)
- [Title page] The title block shows 'TWO PROPER TIES OF OPTIMISERS'; this should read 'TWO PROPERTIES OF OPTIMISERS'.
- [Lemma 3.2, spherical case] The definition τ:=|γ˙|²t is dimensionally inconsistent and the subsequent solution β(t)=a cos τ + b sin τ is hard to follow. It appears τ should be |γ˙|t. Please correct and clarify the reparameterization.
- [Definition 1.1 / RΣ] For hyperbolic λ≤1, RΣ(λ) is not defined, but Theorem 1.2(i) covers this range. The text should explicitly state that the proof for this range uses only the touching-sphere comparison in Corollary 3.3 and does not rely on the lens/rolling-ball arguments.
Circularity Check
No circularity: derivation uses independent stability and rolling-ball results; no fitted parameters or self-citation backbone.
full rationale
The paper's main theorem is proved by contradiction: if a C^2 λ-convex body had maximal area for fixed volume, Lemmas 3.1–3.3 show it contains a strictly λ-convex C^2 patch; Proposition 3.5 then constructs a volume-preserving variation that increases area, contradicting maximality. None of these steps is defined in terms of the conclusion. The variational formulas (Prop 2.2), stability criteria (Prop 2.4, Prop 2.5 = [8, Prop 4.4]), and supporting-ball theorem (Thm 2.6 from [7]/[3]/[6]) are independent external inputs with stated assumptions that do not include the target result. No parameter is fitted to data, and no quantity called a 'prediction' is the fit itself. The only flagged weakness is in hyperbolic space λ≤1: Prop 3.4's stability statement explicitly assumes the constant-mean-curvature piece is not contained in a horosphere, while Corollary 3.3 does not rule out a horosphere patch, so the written proof of Theorem 1.2(i) may have a gap in that case. That is an unproven-case gap, not a circular reduction: the missing horosphere stability computation is not equivalent to the theorem by construction. Hence the appropriate circularity score is 0.
Assumptions & free parameters
assumptions (5)
- standard math First and second variation formulae for volume and area under compactly supported deformations (Prop 2.2).
- standard math Blaschke rolling theorem: a λ-convex body is contained in every supporting ball of radius RΣ(λ) (Thm 2.6).
- standard math Strong stability criterion: a positive u with T(u)≤0 implies strong stability, and a Killing field with ⟨ν,X⟩>0 gives stability (Props 2.4, 2.5).
- standard math Enclosing-sphere comparison: a C² λ-convex body in a ball of radius ρ<RΣ(λ) has a boundary point with smallest principal curvature >λ (Cor 3.3).
- ad hoc to paper Local preservation of λ-convexity under sufficiently small C² deformations supported in a strictly λ-convex patch (Prop 3.5).
Cite this review
Pith. "Pith review of Two properties of optimisers for the reverse isoperimetric problem." pith.science (2026). https://pith.science/paper/J2BARXY4
@misc{pith2026251102688,
author = {Pith},
title = {Pith review of: Two properties of optimisers for the reverse isoperimetric problem},
year = {2026},
howpublished = {\url{https://pith.science/paper/J2BARXY4}},
note = {Machine review of arXiv:2511.02688}
}
abstract
The reverse isoperimetric problem asks for existence and properties of bounded convex sets in a Riemannian manifold which maximise the perimeter under all those sets of fixed volume which roll freely in a ball of some given radius. If the boundary of the set is of class $C^{2}$, this amounts to a positive lower bound on the principal curvatures and in this class we prove that there are no $C^{2}$-maximisers of perimeter with prescribed volume. In addition, we prove that a given possibly non-$C^{2}$ maximiser has its smallest principal curvature constant in regions where it is of class $C^{2}$. We prove this result in the Euclidean, spherical and hyperbolic space.
Reference graph
Works this paper leans on
-
[1]
Henri Anciaux and Brendan Guilfoyle,On the three-dimensional Blascke-Lebesgue problem, Proc. Am. Math. Soc.139(2011), no. 5, 1831–1839
2011
-
[2]
Z.185(1984), no
Joao Barbosa and Manfredo Do Carmo,Stability of hypersurfaces with constant mean curvature, Math. Z.185(1984), no. 3, 339–353
1984
-
[3]
11, 1565–1583
Alexandr Borisenko and Kostiantyn Drach,Closeness to spheres of hypersurfaces with normal curvature bounded below, Sbornik: Math.204(2013), no. 11, 1565–1583
2013
-
[4]
Notes 95(2014), no
,Isoperimetric inequality for curves with curvature bounded below, Math. Notes 95(2014), no. 5, 590–598
2014
-
[5]
Math.353(2019), 431–445
Roman Chernov, Kostiantyn Drach, and Kateryna Tatarko,A sausage body is a unique solution for a reverse isoperimetric problem, Adv. Math.353(2019), 431–445
2019
-
[6]
Kostiantyn Drach,The Blaschke rolling theorem in Riemannian manifolds of bounded curvature, arxiv:2404.02739, 2024
arXiv 2024
-
[7]
Kostiantyn Drach and Kateryna Tatarko,Reverse isoperimetric problems under cur- vature constraints, arxiv:2303.02294, 2023
arXiv 2023
-
[8]
Nachr.297(2024), no
Maria Fernanda Elbert and Barbara Nelli,On the stability of constant higher order mean curvature hypersurfaces in a Riemannian manifold, Math. Nachr.297(2024), no. 11, 4031–4043
2024
Show all 10 references
-
[9]
39, In- ternational Press of Boston Inc., Sommerville, 2006
Claus Gerhardt,Curvature problems, Series in Geometry and Topology, vol. 39, In- ternational Press of Boston Inc., Sommerville, 2006
2006
-
[10]
Robert Osserman,The isoperimetric inequality, Bull. Am. Math. Soc.84(1978), no. 6, 1182–1238. Goethe-Universit¨at Institut f¨ur Mathematik Robert-Mayer-Str. 10 60325 Frankfurt Germany hamdy@stud.uni-frankfurt.de scheuer@math.uni-frankfurt.de
1978
Reviewed August 4, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.