An Isoperimetric inequality for an integral operator on flat tori

TL;DR

This paper proves that among all flat tori with fixed volume, the equilateral torus maximizes the operator and Hilbert-Schmidt norms for a class of integral operators with positive, decreasing, isotropic kernels.

Contribution

It establishes an isoperimetric inequality showing the equilateral torus optimizes certain integral operator norms among all flat tori.

Findings

The equilateral torus maximizes the operator norm.

The equilateral torus maximizes the Hilbert-Schmidt norm.

The result applies to a class of positive, decreasing, isotropic kernels.

Abstract

We consider a class of Hilbert-Schmidt integral operators with an isotropic, stationary kernel acting on square integrable functions defined on flat tori. For any fixed kernel which is positive and decreasing, we show that among all unit-volume flat tori, the equilateral torus maximizes the operator norm and the Hilbert-Schmidt norm.