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Barycentric stability of nonlocal perimeters: the convex case

This paper establishes a sharp nonlocal quantitative isoperimetric inequality involving barycentric asymmetry for convex sets, serving as a nonlocal analogue to Fuglede's 1993 result.

Original authors: Chiara Gambicchia, Enzo Maria Merlino, Berardo Ruffini, Matteo Talluri

Published 2026-01-15

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CC BY 4.0

Original authors: Chiara Gambicchia, Enzo Maria Merlino, Berardo Ruffini, Matteo Talluri

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a lump of clay. In the world of geometry, there is a famous rule: if you want to shape that clay into a form with the smallest possible surface area for a given amount of volume, you must make it a perfect sphere (or a ball in 3D). This is the classic "Isoperimetric Inequality."

But what if your shape isn't a perfect sphere? What if it's slightly squashed, or has a tiny bump? How "wrong" is it?

This paper asks a specific question: If a shape is almost a sphere, how close is its center of gravity (barycenter) to the center of the perfect sphere it should be?

Here is the breakdown of the paper's findings, translated into everyday language:

1. The Problem: Measuring "Wrongness"

Mathematicians have a way to measure how much "extra" surface area a shape has compared to a perfect ball. They call this the deficit. If the deficit is zero, it's a perfect ball. If it's small, the shape is almost a ball.

To measure how far the shape is from being a ball, they usually try to find the best possible ball to compare it to. Imagine sliding a perfect ball around your weird shape until you find the spot where the ball overlaps the most with your shape. The amount of "mismatch" (the parts of the shape outside the ball and the parts of the ball outside the shape) is the standard measure of asymmetry.

The Catch: Finding that "best" ball requires a lot of trial and error (optimization). It's computationally expensive, like trying to find the perfect parking spot by driving around the whole block.

2. The Paper's Idea: The "Barycentric" Shortcut

The authors propose a simpler, "lazy" way to measure this. Instead of searching for the perfect ball, they say: "Just use the ball that is centered exactly where the shape's center of gravity is."

The Analogy: Imagine a seesaw. The center of gravity (barycenter) is the exact spot where you could balance the shape on a single finger. The authors say, "Let's just draw a perfect ball right under that finger."
Why it's tricky: For some weird, disconnected shapes (like two balls far apart), the center of gravity might land in empty space, far away from the actual shape. In those cases, this "lazy" method fails.
The Solution: The authors prove that if the shape is convex (meaning it has no dents, holes, or "C" shapes—it's all solid and bulging outward), this lazy method works perfectly.

3. The "Nonlocal" Twist

The paper doesn't just look at the surface of the shape (like skin). It looks at nonlocal perimeters.

The Analogy: Imagine your shape is a magnet. In the "local" world, only the skin matters. But in this "nonlocal" world, every tiny piece of the shape "feels" every other piece, even if they are far apart, like a long-range gravitational pull. The strength of this pull depends on a parameter called $s$ .
The authors show that even with this weird, long-range interaction, the rule still holds: If a convex shape has very little "extra" nonlocal surface area, then the ball centered at its center of gravity is a very good approximation of the shape.

4. The Main Result (The "Sharp" Inequality)

The paper proves a mathematical "speed limit." It says:

"If you have a convex shape and its 'nonlocal error' (deficit) is tiny, then the distance between your shape and the ball centered at its center of gravity is also tiny—specifically, it shrinks at a predictable rate."

They establish a precise formula showing that the "mismatch" is proportional to the square root of the "error." This is the best possible rate (sharp), meaning you can't get a better estimate.

5. Why Convexity Matters

The authors had to restrict their proof to convex shapes.

The Counter-Example: They explain that if you take a big ball and attach a tiny, tiny speck of clay very far away, the center of gravity will shift slightly toward the speck, but the speck is so small it barely changes the total volume. However, the "mismatch" between the shape and the ball centered at that new gravity point becomes huge (almost the whole shape is now "wrong").
Because convex shapes can't have these "far-away specks" or "dents," the center of gravity stays safely inside the shape, making the math work.

Summary

In short, this paper is a mathematical guarantee for convex shapes interacting with long-range forces. It proves that you don't need to do complex calculations to find the "best" matching ball. You can simply find the shape's center of gravity, draw a ball there, and know that if the shape is nearly perfect, that ball will be a nearly perfect match. This makes calculations much faster and easier for computers and mathematicians.

Technical Summary: Barycentric Stability of Nonlocal Perimeters: The Convex Case

Problem Statement
The paper addresses the quantitative stability of the isoperimetric inequality within the framework of nonlocal (fractional) perimeters. Specifically, it investigates whether a set $E$ that nearly minimizes the fractional $s$ -perimeter must be geometrically close to a ball, measured by the barycentric asymmetry $\lambda_0(E)$ .

While the classical quantitative isoperimetric inequality (relating the isoperimetric deficit to the Fraenkel asymmetry) is well-established for general sets, the version involving barycentric asymmetry is known to hold only under specific geometric constraints (e.g., convexity or boundedness) in the local case ( $s=1$ ). The authors aim to extend the sharp barycentric quantitative inequality to the fractional setting ( $s \in (0,1)$ ) for convex sets. The barycentric asymmetry is defined as the normalized volume of the symmetric difference between the set $E$ and the ball centered at $E$ 's barycenter, $\text{bar}(E)$ , with the same volume. This choice avoids the optimization required in the Fraenkel definition, making it computationally advantageous, though it is a stronger condition that generally requires additional geometric assumptions to hold.

Methodology
The proof follows the strategy established by Fuglede for the local convex case, adapted to the nonlocal setting. The methodology proceeds in three main stages:

Reduction to the Small-Deficit Regime:
The authors first establish that for convex sets with a sufficiently small $s$ -isoperimetric deficit $\delta_s(E)$ , the set must be uniformly bounded. This is achieved via Proposition 3.3, which derives a lower bound for the fractional perimeter in terms of the inradius $r_E$ of a convex set. By combining this with the definition of the deficit, they prove that sets with small deficits cannot have arbitrarily large diameters. This allows the problem to be reduced to sets contained within a fixed cube $Q_l$ .
Continuity and Compactness:
Using the uniform boundedness established above, Lemma 3.1 demonstrates that for sets contained in a fixed cube, a small deficit implies small barycentric asymmetry. This relies on the lower semicontinuity of the fractional perimeter with respect to $L^1$ convergence and the fact that the ball is the unique minimizer.
Stability for Nearly Spherical Sets:
For sets that are close to a ball (nearly spherical), the authors utilize a Fuglede-type spectral estimate (Theorem 2.3). They parametrize the boundary of the set as a perturbation of the unit sphere and relate the deficit $\delta_s(E)$ to the $L^2$ norm of the perturbation function $u$ . By combining this spectral lower bound with the geometric relationship between the $L^1$ norm of $u$ and the barycentric asymmetry, they derive the desired inequality for the "small deficit" case.
Global Argument:
The final proof (Theorem 1.1) combines the two regimes. If the deficit is large, the inequality holds trivially due to the boundedness of the asymmetry. If the deficit is small, the set is shown to be nearly spherical (via the diameter bound and Lemma 3.1), allowing the application of the spectral estimate.

Key Contributions and Results
The primary result is Theorem 1.1, which establishes a sharp quantitative isoperimetric inequality for convex sets in the fractional setting:
$\lambda_0(E) \leq C \sqrt{\delta_s(E)}$
where $C$ depends only on the dimension $n$ and the fractional parameter $s$ .

Key technical components include:

Inradius Estimate: The derivation of a lower bound for the fractional perimeter of a convex set in terms of its inradius (Equation 3.4), specifically $P_s(E) \geq \alpha_{n,s} |E| / r_E^s$ . This estimate is noted as potentially of independent interest.
Extension of Fuglede's Result: The paper successfully adapts the local convex stability result to the nonlocal fractional perimeter, confirming that the exponent $1/2$ remains sharp for the barycentric asymmetry in this context.
Limit Behavior: The authors show in Remark 4.2 that their result is consistent with the local case ( $s \to 1$ ) and can be used to recover the local barycentric inequality via limit arguments involving the convergence of fractional deficits to the classical deficit.

Significance and Claims
The authors position this work as the initiation of the study of barycentric quantitative isoperimetric inequalities in the fractional setting. They explicitly state that while the proof is currently restricted to convex sets, the methodology is robust enough to extend to "nearly spherical sets" (Remark 4.1), which is a crucial step toward broader applicability.

The paper emphasizes that, similar to the local case, the barycentric inequality cannot hold for general sets without additional assumptions (such as equi-boundedness), citing a counterexample involving two separated balls. The work serves as a nonlocal analogue to Fuglede's seminal result [16], bridging the gap between local geometric stability and nonlocal variational problems. The authors do not claim to solve the problem for general non-convex sets but rather establish the sharp result for the convex class, providing a foundation for future extensions.