The anisotropic fractional isoperimetric problem with respect to unconditional unit balls

The minimizers of the anisotropic fractional isoperimetric inequality with respect to the convex body $K$ in ${\mathbb{R}}^n$ are shown to be equivalent to star bodies whenever $K$ is strictly convex and unconditional. From this a P'olya-Szeg"o principle for anisotropic fractional seminorms is derived by using symmetrization with respect to star bodies.