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Optimisation de forme dans la classe des corps de largeur constante et des rotors.

Abstract : In this dissertation, we study the minimization of a geometrical functional in dimension 2 and 3 under boundary constraints. In the second part, we study constant width bodies in dimension 2 and we prove the Blaschke-Lebesgue's theorem by the optimal control theory (Pontryagin maximum principle). In dimension 3, we study the problem of minimizing the volume in the class of constant width bodies which have an axis of revolution. By Pontryagin's principle, we derive necessary conditions satisfied by a minimizer. In the third part, we study the problem of minimizing the area in the class of rotors. By Pontryagin's principle, we show that the boundary of a minimizer is a finite union of arcs of circle whose radii have prescribed values. In the fourth part, we investigate optimal locality properties of the area for regular rotors under a certain type of perturbations. Thanks to Kuhn-Tucker's theorem, we generalize a result of Firey in the case of rotors. We thus show that the regular rotors of the equilateral triangle are local maximizers of the area, whereas the rotors in a regular polygon with n>4 sides are saddle points. In the fifth part, we study the problem of minimizing the volume among constant width bodies in dimension 3. We present a complete analytic parametrization of the constant width bodies. We derive weak optimality conditions for a minimizer of the functional.
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Contributor : Térence Bayen <>
Submitted on : Tuesday, January 22, 2008 - 2:05:58 PM
Last modification on : Wednesday, December 9, 2020 - 3:09:15 PM
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  • HAL Id : tel-00212070, version 1

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Térence Bayen. Optimisation de forme dans la classe des corps de largeur constante et des rotors.. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2007. Français. ⟨tel-00212070⟩

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