A sharp stability result for the relative isoperimetric inequality inside convex cones

The relative isoperimetric inequality inside an open, convex cone C states that, at fixed volume, Br ∩ C minimizes the perimeter inside C. Starting from the observation that this result can be recovered as a corollary of the anisotropic isoperimetric inequality, we exploit a variant of Gromov’s proof of the classical isoperimetric inequality to prove a sharp stability result for the relative isoperimetric inequality inside C. Our proof follows the line of reasoning in [16], though several new ideas are needed in order to deal with the lack of translation invariance in our problem