arxiv: 2605.07543 · v1 · submitted 2026-05-08 · 🧮 math.AP

Recognition: no theorem link

Stability of the ball in isoperimetric inequalities between two fractional perimeters

A. Massaccesi, G. Alberti, G. Cozzi, J. Mirmina

Pith reviewed 2026-05-11 02:47 UTC · model grok-4.3

classification 🧮 math.AP

keywords fractional perimetersisoperimetric ratiostabilitynearly spherical setsSobolev normlocal minimizer

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The pith

The ball is a local minimizer of the scale-invariant isoperimetric ratio between two fractional perimeters among nearly spherical sets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to prove that the ball minimizes the functional F which takes the t-perimeter to the power 1 over n minus t divided by the s-perimeter to the power 1 over n minus s, for sets that are close to being balls. A sympathetic reader would care because this shows the ball's optimality persists when both terms in the isoperimetric ratio are fractional perimeters rather than one being volume. The proof proceeds by parametrizing nearly spherical sets with a small function u on the sphere and showing that F increases when u is nonzero in a Sobolev sense. This provides quantitative control on the deviation from the ball.

Core claim

The ball is a local minimizer of the scale-invariant isoperimetric ratio F(E) = P_t(E)^{1/(n-t)} / P_s(E)^{1/(n-s)} among nearly spherical sets E. This is shown by rewriting F as a functional of the boundary parametrization u on the unit sphere and establishing quantitative stability around the zero function in a suitable Sobolev norm. The result parallels known stability when the s-perimeter is replaced by the volume.

What carries the argument

The functional of the boundary perturbation u on the sphere whose variation is controlled in the Sobolev norm to establish the local minimum at the ball.

If this is right

The ratio F increases at least quadratically with the Sobolev norm of the perturbation u.
The minimality holds locally for any choice of 0 < s < t < 1.
This stability result can be used to study the behavior of the isoperimetric ratio under small deformations of the ball.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same perturbation method might apply to prove stability in other inequalities involving multiple fractional operators.
One could test the result by computing the ratio for explicit perturbations like ellipsoids or spherical harmonics expansions of u.
If the local stability extends globally, it would imply the ball is the unique minimizer in a larger class of sets.

Load-bearing premise

The sets under consideration have boundaries that can be expressed as small perturbations of the sphere via a scalar function u.

What would settle it

Computing the value of the ratio F for a specific nearly spherical set with a nonzero perturbation u and finding it smaller than for the ball would falsify the claim.

read the original abstract

We consider the isoperimetric inequality involving the $s$-perimeter and the $t$-perimeter with $0<s<t<1$, and show that the ball is a local minimizer of the (scale-invariant) isoperimetric ratio $\mathcal{F}(E):=P_t(E)^{\frac{1}{n-t}}/ P_s(E)^{\frac{1}{n-s}}$ among sets $E$ that are nearly spherical. To this end, we rewrite $\mathcal{F}$ as a functional of $u$, where $u$ is a scalar function on the unit sphere in $\mathbb{R}^n$ that parametrizes the boundary of $E$, and prove a quantitative stability result for $\mathcal{F}$ around $u=0$ with respect to a suitable Sobolev norm. This parallels known results where the $s$-perimeter is replaced by the volume.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends local stability to the ratio of two fractional perimeters via the standard nearly-spherical reduction and second-variation analysis.

read the letter

The main result is that the ball is a local minimizer of the scale-invariant ratio F(E) = P_t(E)^{1/(n-t)} / P_s(E)^{1/(n-s)} among nearly spherical sets, for 0 < s < t < 1. The authors parametrize the boundary by a small function u on the sphere, rewrite F in terms of u, and analyze its second variation using the known spectral expansions of the two fractional perimeters. After removing constants to account for scale invariance, the resulting quadratic form is coercive in the appropriate Sobolev space, which directly yields the quantitative stability estimate around u = 0.

Referee Report

2 major / 3 minor

Summary. The paper proves that the unit ball is a local minimizer of the scale-invariant ratio F(E) = P_t(E)^{1/(n-t)} / P_s(E)^{1/(n-s)} for 0 < s < t < 1 among nearly spherical sets E in R^n. The proof parametrizes the boundary of such sets by a small scalar function u on the unit sphere S^{n-1}, rewrites F as a functional of u, verifies that the first variation vanishes at u=0 by rotational symmetry, derives the associated quadratic form for the second variation by combining the known spectral expansions of the s- and t-perimeters (via the fractional Laplacian on the sphere), and establishes coercivity of this quadratic form in the appropriate Sobolev space (orthogonal to constants) to obtain a quantitative stability estimate.

Significance. If the second-variation calculation holds, the result extends existing stability theorems for single fractional perimeters (or volume) to the ratio of two distinct fractional perimeters. This provides a concrete quantitative control near the ball that could serve as a starting point for global stability or for studying competition between nonlocal perimeters. The approach is direct and avoids circularity by relying on explicit spectral expansions rather than fitted quantities.

major comments (2)

[§3.2] §3.2, around the expansion of the second variation: the quadratic form obtained by differentiating the ratio F twice must be shown to remain positive definite after the normalization that accounts for scale invariance. The manuscript combines the known eigenvalues for P_s and P_t, but the explicit verification that the resulting coefficients stay positive for all 0 < s < t < 1 (without sign-indefinite cross terms) is only sketched; a short lemma isolating the lowest spherical-harmonic modes would make the coercivity argument fully transparent.
[§4] §4, quantitative stability statement: the passage from the coercive quadratic form to the L^2 or H^{(n-s)/2} stability estimate for ||u|| relies on a standard Taylor expansion with remainder. The constant in the inequality is stated to depend only on n,s,t, but the dependence on the distance to the ball (i.e., the size of the neighborhood in which the estimate holds) is not quantified; this is needed to confirm that the local minimizer property is uniform in the stated topology.

minor comments (3)

[§2] The notation for the fractional perimeters P_s and P_t is introduced in §1 but the precise normalization constants (e.g., the factor 2 in the double-integral definition) are not repeated when the functional F is rewritten in terms of u; adding a short reminder would help the reader follow the differentiation.
[Figure 1] Figure 1 (schematic of nearly-spherical sets) is clear, but the caption should explicitly state the range of the smallness parameter ||u||_{H^1} for which the parametrization is valid.
[References] A few typographical inconsistencies appear in the bibliography (e.g., missing volume numbers for some fractional-perimeter references); these are easily corrected.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise suggestions for improvement. We address each major comment below and will revise the manuscript accordingly to enhance transparency.

read point-by-point responses

Referee: [§3.2] §3.2, around the expansion of the second variation: the quadratic form obtained by differentiating the ratio F twice must be shown to remain positive definite after the normalization that accounts for scale invariance. The manuscript combines the known eigenvalues for P_s and P_t, but the explicit verification that the resulting coefficients stay positive for all 0 < s < t < 1 (without sign-indefinite cross terms) is only sketched; a short lemma isolating the lowest spherical-harmonic modes would make the coercivity argument fully transparent.

Authors: We thank the referee for highlighting this point. The quadratic form for the second variation of F is obtained by combining the spectral expansions of the s- and t-perimeters; positivity follows because the eigenvalues of the fractional Laplacian on the sphere are strictly increasing in the fractional order, so the weighted difference remains positive on the space orthogonal to constants (i.e., for all spherical harmonics of degree l ≥ 1). To render the argument fully explicit and transparent, we will add a short lemma that isolates the lowest modes (starting from l=1) and verifies that the resulting coefficients are positive for every 0 < s < t < 1. This lemma will be inserted in §3.2 of the revised manuscript. revision: yes
Referee: [§4] §4, quantitative stability statement: the passage from the coercive quadratic form to the L^2 or H^{(n-s)/2} stability estimate for ||u|| relies on a standard Taylor expansion with remainder. The constant in the inequality is stated to depend only on n,s,t, but the dependence on the distance to the ball (i.e., the size of the neighborhood in which the estimate holds) is not quantified; this is needed to confirm that the local minimizer property is uniform in the stated topology.

Authors: We agree that an explicit reference to the size of the neighborhood improves the statement. The Taylor expansion of F around u=0 with integral remainder yields the desired quantitative inequality as soon as ||u|| is smaller than a threshold δ>0 whose existence is guaranteed by the continuity of the higher-order terms; this δ depends only on n, s and t. In the revised version we will explicitly record that there exists δ=δ(n,s,t)>0 such that the stability estimate holds for all admissible u with ||u||_{H^{(n-s)/2}(S^{n-1})} < δ, thereby confirming uniformity of the local minimizer property in the stated topology. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds by parametrizing nearly spherical sets via a small scalar function u on the sphere, rewriting the scale-invariant ratio F as a functional of u, verifying that the first variation vanishes at u=0 by rotational symmetry, and then computing the second variation quadratic form from the known spectral expansions of the fractional perimeters (via the fractional Laplacian on the sphere). Coercivity of this quadratic form in the appropriate Sobolev space is established directly from the spectral properties for 0 < s < t < 1, yielding the quantitative stability estimate. All steps rely on standard analytic techniques and externally known spectral facts rather than any fitted parameter, self-referential definition, or load-bearing self-citation chain; the central claim therefore remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available. The proof relies on standard properties of fractional perimeters, the sphere, and Sobolev spaces; no free parameters, ad-hoc axioms, or invented entities are visible from the summary.

axioms (1)

standard math Standard functional-analytic properties of fractional perimeters and Sobolev norms on the sphere
Invoked when rewriting F as a functional of u and analyzing its stability.

pith-pipeline@v0.9.0 · 5458 in / 1285 out tokens · 54888 ms · 2026-05-11T02:47:07.065351+00:00 · methodology

discussion (0)

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