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Journal of Convex Analysis 16 (2009), No. 2, 391--407
Copyright Heldermann Verlag 2009

An Optimal Control Theory Approach to the Blaschke-Lebesgue 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

According to the Blaschke-Lebesgue 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 non-negative and bounded above, so that the Blaschke-Lebesgue 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 3-dimensional bodies of revolution having constant width.

Keywords: Blaschke-Lebesgue theorem, control theory.

MSC: 52A40; 49Q10, 52A15, 52A38

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