The ball locally minimizes F(E) = P_t(E)^{1/(n-t)} / P_s(E)^{1/(n-s)} (0 < s < t < 1) among nearly spherical sets, via quantitative stability of the rewritten functional in a Sobolev norm on the sphere.
A selection principle for the sharp quantitative isoperi- metric inequality
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Stability of the ball in isoperimetric inequalities between two fractional perimeters
The ball locally minimizes F(E) = P_t(E)^{1/(n-t)} / P_s(E)^{1/(n-s)} (0 < s < t < 1) among nearly spherical sets, via quantitative stability of the rewritten functional in a Sobolev norm on the sphere.