Geodesic
An Optimal Control Theory Approach to the Blaschke-Lebesgue Theorem
Journal of convex analysis, Tome 16 (2009) no. 2, pp. 391-407.

Voir la notice de l'article provenant de la source Heldermann Verlag

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.
Classification : 52A40, 49Q10, 52A15, 52A38
Mots-clés : Blaschke-Lebesgue theorem, control theory 
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     title = {An {Optimal} {Control} {Theory} {Approach} to the {Blaschke-Lebesgue} {Theorem}},
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     number = {2},
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F. Malagoli. An Optimal Control Theory Approach to the Blaschke-Lebesgue Theorem. Journal of convex analysis, Tome 16 (2009) no. 2, pp. 391-407. http://geodesic.mathdoc.fr/item/JCA_2009_16_2_JCA_2009_16_2_a4/