 If we define the 1-form formula_51, then we can rewrite this lower bound using the Stokes formula as

The boundary integral that appears here is defined in terms of the distance functions formula_30 restricted to the boundary, which "do not depend on the isometric filling".The result of the integral therefore depends only on the placement of the points formula_26 on the circle of length "2L".We omitted the computation, and expressed the result in terms of the lengths formula_55 of each counterclockwise boundary arc from a point formula_26 to the following point formula_57.