  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.
