Loewner's Inequality With A Defect Term
The explanation of the strengthened version of Loewner's inequality is somewhat more technical than the rest of this article. It seems worth including it here for the sake of completeness. The strengthened version is the inequality
where Var is the probabilistic variance while f is the conformal factor expressing the metric g in terms of the flat metric of unit area in the conformal class of g. The proof results from a combination of the computational formula for the variance and Fubini's theorem (see Horowitz et al, 2009).
Read more about this topic: Introduction To Systolic Geometry
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