The anisotropic fractional isoperimetric problem with respect to unconditional unit balls

Fractional perimeters on the sphere

“…General references for the discussion and proof of 2 and 3 are [22] and [12] respectively. There are nowadays a lot of isoperimetric inequalities in different settings, for example for the Gauss measure (see [4], [25]), on the sphere (first discovered by Paul Lévy) and for anisotropic fractional perimeters (see [15], [17]). We apply a general rearrangement inequality established by Beckner [2] to spherical fractional perimeters and derive that for every spherical cap C ⊆ S n and every Borel set E with H n (E) = H n (C) and every s ∈ (−n, 1) we have…”