Sharp quantitative stability for isoperimetric inequalities with homogeneous weights

We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimet-ric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequalities through the ABP method (see X. Cabre, X. Ros-Oton, and J. Serra [J. Eur. Math. Soc. (JEMS) 18 (2016), pp. 2971-2998]), we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set E and the minimizer of the inequality (as in Gromov's proof of the isoperimetric inequal-ity). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of [Figalli, Maggi, and Pratelli [Invent. Math. 182 (2010), pp. 167-211] and prove that if E is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations. As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest.