  A version of the Riesz-Sobolev convolution inequality is formulated and
proved for arbitrary compact connected Abelian groups. Maximizers are
characterized and a quantitative stability theorem is proved, under natural
hypotheses. A corresponding stability theorem for sets whose sumset has nearly
minimal measure is also proved, sharpening recent results of other authors. For
the special case of the group $\mathbb{R}/\mathbb{Z}$, a continuous deformation
of sets is developed, under which an appropriately scaled Riesz-Sobolev
functional is shown to be nondecreasing.
