An improved quantitative fractional isoperimetric inequality

Abstract: 

We present an improved version of the quantitative fractional isoperimetric inequality, in which a stronger notion of asymmetry appears. In particular, we show that the square root of the isoperimetric deficit controls, not only the Fraenkel asymmetry, but also a suitable notion of oscillation of the boundary. This extends the result in the local setting obtained by Fusco and Julin in 2011. The proof follows a similar spirit but is different with respect to that in the local case. This is a joint work with E. Cinti and E. M. Merlino.