Barycentric Stability of Nonlocal Perimeters: the Convex Case

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 B. Fuglede [Lower estimate of the isoperimetric deficit of convex domains in Rⁿ in terms of asymmetry, Geom. Dedicata 47/1 (1993) 41–48]. A main tool in the proof is an estimate from below of the fractional perimeter by a negative power of the inradius for convex sets