Geodesic
Optimal Sets for a Class of Minimization Problems with Convex Constraints
Journal of convex analysis, Tome 19 (2012) no. 3, pp. 725-758.

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

We look for the minimizers of the functional Jλ(Ω) = λ|Ω| - P(Ω) among planar convex domains constrained to lie into a given ring. We prove that, according to the values of the parameter λ, the solutions are either a disc or a polygon. In this last case, we describe completely the polygonal solutions by reducing the problem to a finite dimensional optimization problem. We recover classical inequalities for convex sets involving area, perimeter and inradius or circumradius and find a new one.
Classification : 52A10, 52A38, 52A40, 49Q10
Mots-clés : Convex geometry, shape optimization, isoperimetric inequalities, length, area 
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     author = {C. Bianchini and A. Henrot},
     title = {Optimal {Sets} for a {Class} of {Minimization} {Problems} with {Convex} {Constraints}},
     journal = {Journal of convex analysis},
     pages = {725--758},
     publisher = {mathdoc},
     volume = {19},
     number = {3},
     year = {2012},
     url = {http://geodesic.mathdoc.fr/item/JCA_2012_19_3_JCA_2012_19_3_a6/}
}
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C. Bianchini; A. Henrot. Optimal Sets for a Class of Minimization Problems with Convex Constraints. Journal of convex analysis, Tome 19 (2012) no. 3, pp. 725-758. http://geodesic.mathdoc.fr/item/JCA_2012_19_3_JCA_2012_19_3_a6/