  An inequality of Brascamp-Lieb-Luttinger and of Rogers states that among
subsets of Euclidean space $\mathbb{R}^d$ of specified Lebesgue measures, 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 $\text{Sl}(d)$. We investigate
functionals of this type, and their maximizers, in perhaps the simplest
situation in which $\text{Sl}(d)$ invariance does not hold. Assuming a more
limited symmetry involving 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 $\text{Sl}(d)$--invariant framework we show that these
distinguished maximizers are strongly convex sets with infinitely
differentiable boundaries. It is shown that maximizers fail to exist for
certain arbitrarily small perturbations of $\text{Sl}(d)$--invariant
structures.
