A strong quantitative form of the fractional isoperimetric inequality
Eleonora Cinti, Enzo Maria Merlino, Berardo Ruffini
TL;DR
The paper extends the sharp quantitative isoperimetric framework to the fractional $s$-perimeter, showing that the fractional deficit $\delta_s(E)$ controls not only the Fraenkel asymmetry but also a boundary oscillation through the index $A_s$. The authors develop a regularization scheme via the energy $P_s(E)+\varepsilon V_s(E)$ to prove ball rigidity and derive a strong stability inequality linking $A_s(E)^2$ to $\delta_s(E)$, with consistent behavior as $s\to1^-$. They also obtain a quantitative fractional Cheeger inequality and identify fundamental limits: stability bounds involving the oscillation index $\beta_s$ fail in general, as demonstrated by explicit counterexamples. The work thus provides a robust nonlocal analogue of Fuglede–Fusco–Julin-type stability, with implications for nonlocal isoperimetric problems and fractional Cheeger-type variational problems. It also connects the $A_s$ index to fractional Sobolev norms near the sphere, highlighting the role of the full nonlocal structure in stability estimates.
Abstract
We show a strong version of the fractional quantitative isoperimetric inequality, in which the isoperimetric deficit controls not only the Fraenkel asymmetry but also a sort of oscillation of the boundary. This generalizes the local result by Fusco and Julin in \cite{FJ}. The proof follows a regularization process as in \cite{FJ} but it is quite different in its spirit. Then, as a consequence of the quantitative inequality, we prove some stability estimates for a fractional Cheeger inequality.