Inequalities with full affine invariance are rare. Fundamental examples include the inequalities of Brunn-Minkowski, Young, Riesz-Sobolev, and Hausdorff-Young. For each of these, a sharp form with an optimal constant is known, including a characterization of all extremizing functions, or sets. This course will discuss refinements of some of these sharp inequalities. These refinements quantify the uniqueness of extremizers. The first lecture will be a general introduction, reviewing several inequalities, stating refinements, and introducing associated ideas. The second lecture will outline a proof of a sharpened Riesz-Sobolev inequality. In contrast to earlier work of the speaker which focused on the exploitation of ideas from (finite) additive combinatorics, this proof is rooted firmly in the continuum. It emphasizes the role of the affine group as a symmetry group of the set of cosets of one-parameter subgroups of the Euclidean group.