Minimality of polytopes in a nonlocal anisotropic isoperimetric problem

We consider the minimization of an energy functional given by sum a crystalline perimeter and nonlocal interaction Riesz type, under volume constraint. show that, in small mass regime, if Wulff shape anisotropic has certain symmetry properties, then it is unique global minimizer total energy. In dimension two this applies to convex polygons which are reflection symmetric with respect bisectors angles. further prove rigidity result for structure (local) minimizers dimensions.