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.