A Sharp Stability Result for the Relative Isoperimetric Inequality Inside Convex Cones

The relative isoperimetric inequality inside an open, convex cone states that, at fixed volume, minimizes the perimeter inside . 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 . Our proof follows the line of reasoning in Figalli et al.: Invent. Math. 182:167-211 (2010), though several new ideas are needed in order to deal with the lack of translation invariance in our problem.