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.

中文翻译：

---

失去对称性的对称化不等式

Brascamp-Lieb-Luttinger 和 Rogers 的不等式表明，在指定 Lebesgue 测度的欧几里得空间 $\mathbb{R}^d$ 的子集中，以原点为中心的球是由多维积分定义的某些泛函的最大化者。对于 $d>1$，这个不等式只适用于在 $\text{Sl}(d)$ 对角线作用下不变的泛函。我们在 $\text{Sl}(d)$ 不变性不成立的最简单情况下研究了这种类型的泛函及其最大化器。假设涉及膨胀而不是旋转的更有限的对称性，我们在自然假设下表明存在最大化器，而且存在其结构反映这种有限对称性的杰出最大化器。对于 $\text{Sl}(d)$--invariant 框架的小扰动，我们表明这些显着的最大化器是具有无限微分边界的强凸集。结果表明，对于$\text{Sl}(d)$--不变结构的某些任意小的扰动，最大化器不存在。