Barycentric stability of nonlocal perimeters: the convex case

TL;DR

The paper proves a sharp nonlocal barycentric stability result for convex sets: for every $s\in(0,1)$ and convex $E$ with finite measure, there exists $C(n,s)$ such that $\lambda_0(E)\le C(n,s)\sqrt{\delta_s(E)}$, where $P_s$ is the fractional perimeter and $\delta_s(E)=\frac{P_s(E)-P_s(B(m))}{P_s(B(m))}$. The approach combines a reduction to the small-deficit regime via compactness, a universal diameter bound for sets with $\delta_s(E)\le1$, and a Fuglede-type near-spherical estimate to obtain the sharp nonlocal stability inequality. The authors also explain how the result extends to nearly spherical sets and discuss necessary geometric assumptions to avoid counterexamples, situating the work as a nonlocal analogue of Fuglede's classical result. The work thus provides a quantitative tool linking nonlocal perimeter deficits to barycentric proximity to balls in the convex setting, with potential implications for nonlocal isoperimetric problems and numerical approximations.

Abstract

In this work, we establish a sharp form of a nonlocal quantitative isoperimetric inequality involving the barycentric asymmetry for convex sets. This result can be seen as the nonlocal analogue of the one obtained by Fuglede in 1993.

Figures (1)

• Figure 1: The angle $x\hat{p}y$ is acute for any $y\in E$.

Theorems & Definitions (12)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Lemma 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof
• proof : Proof of Theorem \ref{['thm:main']}