A generalization of the isoperimetric inequality

Abstract

For a simple closed plane curve of length L bounding an area A the classical isoperimetric inequality asserts that U -A > 0 , with equality holding only for a circle. We show here that this inequality remains true for non-simple closed curves where in place of A we take the sum of the areas into which the curve divides the plane, each weighted with the square of the winding number, i.e., -/ w 2 dA > 0 where, for p e E\ w(p) is the winding number of p with respect to the curve. Equality holds if and only if the curve is a circle, or a circle traversed several times or several coincident circles each traversed in the same direction any number of times. Note that this implies that U -4 C\w\*dA > 0 for any 0 < p < 2 and that 2 is here the best possible power.