T. Bonnesen [Math. Ann. 91, 252-268 (1924)] outlined a proof that for a closed convex curve K there is a unique annulus which bi-encloses K. For this annulus he established the inequality \((R-r)^ 2\leq (L^ 2/4\pi)-A,\) where L denotes the length of K, A the area enclosed by K and R and r are the radii of the bi-enclosing annulus. The author extends Bonnesen’s proof to the case of nonconvex simply closed curves.
First he introduces the notion of a bi-enclosing annulus for a closed Jordan curve. Then he proves that for any simply closed curve K in the plane there exists an annulus which bi-encloses K. As a next step for the proof of his main result he shows that for every rectifiable closed curve K (simple or not) Bonnesen’s inequality holds for any annulus which bi- encloses K. Finally combining both results he obtains the proposed generalization of this inequality to the described class of nonconvex curves.