A quantitative weighted isoperimetric inequality via the ABP method

We prove a quantitative version for a weighted isoperimetric inequality in cones (recently proven by Cabré, Ros, Serra), for weights which satisfies some concavity conditions in the cone. The proof uses the ABP method applied to solutions of a certain Neumann problem, combined with a weighted trace inequality, which let us the get the optimal exponent on the isoperimetric deficit. This is a joint work with X. Cabre’, A. Pratelli, X. Ros and J. Serra.