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

The relative isoperimetric inequality inside an open, convex cone $\mathcal C$ states that, at fixed volume, $B_r \cap \mathcal C$ 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.