Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Blaschke-Lebesgue theorem
  • control theory

Mathematics Subject Classification

52A40; 49Q10, 52A15, 52A38

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Suggested citation

F. Malagoli. “An Optimal Control Theory Approach to the Blaschke-Lebesgue Theorem.” Journal of Convex Analysis 16 (2009), No. 2, 391–407. https://doi.org/10.68381/jca16021

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.