  For a domain $\Omega$ in a geodesically convex surface, we introduce a
scattering energy $\mathcal{E}(\Omega)$, which measures the asymmetry of
$\Omega$ by quantifying its incompatibility with an isometric circle action. We
prove several sharp quantitative isoperimetric inequalities involving
$\mathcal{E}(\Omega)$ and characterize the domains with vanishing scattering
energy by their convexity and rotational symmetry. We also give a new proof of
the sharp Sobolev inequality for Riemannian surfaces which is independent of
the isoperimetric inequality.
