  An inequality of Brascamp-Lieb-Luttinger generalizes the Riesz-Sobolev
inequality, stating that certain multilinear functionals, acting on nonnegative
functions of one real variable with prescribed distribution functions, are
maximized when these functions are symmetrized. It is shown that under certain
hypotheses, when the functions are indicator functions of sets of prescribed
measures, then up to the natural translation symmetries of the inequality, the
maximum is attained only by intervals centered at the origin. Moreover, a
quantitative form of this uniqueness is established, sharpening the inequality.
The hypotheses include an auxiliary genericity assumption which may not be
necessary.
