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

The relative isoperimetric inequality inside an open, convex cone $\mathcal C$statesthat,atfixedvolume,$B_r \cap \mathcal C$ minimizes the perimeter inside $\mathcal{C}$. 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 $\mathcal{C}$. Our proof follows the line of reasoning in \cite{Fi}, though several new ideas are needed in order to deal with the lack of translation invariance in our problem.