An application of the maximum principle to the geometry of plane curves

The maximum principle of control theory is used to find necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length. This result is used to prove that such a closed curve having length L L and curvature k k satisfying | k | ≦ K |k| K can be contained in a circle of radius R R , where R ≦ L / 4 − ( π − 2 ) / 2 K R L/4 - (π - 2)/2K .