Bonnesen's inequality for the isoperimetric deficiency of closed curves in the plane

An inequality of T. Bonnesen for the isoperimetric deficiency of a convex closed curve in the plane is extended to arbitrary simple closed curves. As a primary tool it is shown that, for any such curve, there exist two concentric circles such that the curve is between these and passes at least four times between them. Next, it is shown that Bonnesen's inequality holds for any such pair of concentric circles.This work was supported by The Danish Natural Science Council and carried out during the author's stay 1989–90 as a Visiting Member at the Institute for Advanced Study, Princeton, N.J., U.S.A.