A symmetrization inequality shorn of symmetry

TL;DR

This paper explores the maximizers of certain geometric functionals under limited symmetry conditions, establishing existence, structure, and regularity of maximizers, and identifying cases where maximizers do not exist.

Contribution

It extends symmetry-based inequalities to settings with limited symmetry, demonstrating existence and properties of maximizers under these conditions.

Findings

Maximizers exist under dilation symmetry assumptions.

Maximizers are strongly convex with smooth boundaries for small perturbations.

Maximizers may fail to exist under certain perturbations 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, 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.