
Journal of Convex Analysis 16 (2009), No. 2, 391407 Copyright Heldermann Verlag 2009 An Optimal Control Theory Approach to the BlaschkeLebesgue Theorem Federica Malagoli Department of Pure and Applied Mathematics "G. Vitali", Univ. of Modena and Reggio Emilia, Via G. Campi 213/B, 41100 Modena, Italy fmalagoli@mail.unimore.it According to the BlaschkeLebesgue theorem, among all plane convex bodies of given constant width the Reuleaux triangle has the least area. The area of a convex set can be written as an integral involving the support function h and the radius of curvature ρ of the set. The support function satisfies a second order ordinary differential equation where the datum is the radius of curvature. The function ρ is nonnegative and bounded above, so that the BlaschkeLebesgue theorem can be formulated as an optimal control problem, where the functional to be minimized is the area. In the same way, the control theory can be used to find the body of minimum volume among all 3dimensional bodies of revolution having constant width. Keywords: BlaschkeLebesgue theorem, control theory. MSC: 52A40; 49Q10, 52A15, 52A38 [ Fulltextpdf (683 KB)] for subscribers only. 