Quantitative isoperimetric inequalities in capillarity problems and cones in strong and barycentric forms

We study quantitative isoperimetric inequalities for two different perimeter-type functionals. We first consider classical capillarity functionals, which measure the perimeter of sets in a Euclidean half-space, assigning a constant weight $λ\in (-1, 1)$ to the portion of the boundary that lies on the boundary of the half-space. We then consider the relative perimeter of sets contained in some suitable convex cone in the Euclidean space. In both settings, we establish sharp quantitative isoperimetric inequalities in the so-called strong form. More precisely, we show that the isoperimetric deficit of a competitor not only controls the Fraenkel asymmetry, but it also controls an oscillation asymmetry that measures how much the unit normals to the boundary of a competitor deviate from those of an isoperimetric set. Our technique is also able to explicitly identify a center that can be employed to compute the asymmetries. In particular, we also derive barycentric versions of these quantitative isoperimetric inequalities (both in the classical and in the strong form). The proofs are based on the derivation of Fuglede-type estimates for graphs defined on general spherically convex domains, in combination with a new application of the selection principle that directly provides the inequalities in barycentric form.