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

The anisotropic fractional isoperimetric problem with respect to unconditional unit balls