Loewner's torus inequality with isosystolic defect

TL;DR

This paper establishes a systolic analog of Bonnesen's isoperimetric defect for Loewner's torus inequality, linking geometric inequalities with probabilistic variance of conformal factors.

Contribution

It introduces a novel isosystolic defect expressed via probabilistic variance, extending classical isoperimetric concepts to systolic geometry.

Findings

Isosystolic defect expressed in terms of variance of conformal factor

Analog of Bonnesen's defect for Loewner's inequality

Connection between systolic geometry and probabilistic measures

Abstract

We show that Bonnesen's isoperimetic defect has a systolic analog for Loewner's torus inequality. The isosystolic defect is expressed in terms of the probabilistic variance of the conformal factor of the metric g with respect to the flat metric of unit area in the conformal class of g.