A symmetrization inequality shorn of symmetry

Abstract:

An inequality of Brascamp-Lieb-Luttinger and of Rogers states that among subsets of Euclidean space $\mathbb {R}^d$ of specified Lebesgue measures, (tuples of) balls centered at the origin are maximizers of certain functionals defined by multidimensional integrals. For $d>1$, this inequality only applies to functionals invariant under a diagonal action of $\operatorname {Sl}(d)$. We investigate functionals of this type, and their maximizers, in perhaps the simplest situation in which $\operatorname {Sl}(d)$ invariance does not hold. Assuming a more limited symmetry encompassing dilations but not rotations, we show under natural hypotheses that maximizers exist, and, moreover, that there exist distinguished maximizers whose structure reflects this limited symmetry. For small perturbations of the $\operatorname {Sl}(d)$–invariant framework we show that these distinguished maximizers are strongly convex sets with infinitely differentiable boundaries. It is shown that in the absence of partial symmetry, maximizers fail to exist for certain arbitrarily small perturbations of $\operatorname {Sl}(d)$–invariant structures.