Loewner's torus inequality with isosystolic defect
  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. 
Charles Horowitz, Karin Usadi Katz, and Mikhail G. Katz