*Accepted Paper*

**Inserted:** 14 sep 2011

**Journal:** J. Geom. Anal.

**Year:** 2011

**Abstract:**

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 a recent paper of Figalli-Maggi-Pratelli, though several new ideas are needed in order to deal with the lack of translation invariance in our problem.

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