Journal of Convex Analysis 19 (2012), No. 3, 725--758
Copyright Heldermann Verlag 2012
Optimal Sets for a Class of Minimization Problems with Convex Constraints
Chiara Bianchini
Institut Elie Cartan, Université Henri Poincaré, Blvd des Aiguillettes, B.P. 70239, 54506 Vandoeuvre-les-Nancy, France
chiara.bianchini@iecn.u-nancy.fr
Antoine Henrot
Institut Elie Cartan, Université Henri Poincaré, Blvd des Aiguillettes, B.P. 70239, 54506 Vandoeuvre-les-Nancy, France
antoine.henrot@iecn.u-nancy.fr
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.
Keywords: Convex geometry, shape optimization, isoperimetric inequalities, length, area.
MSC: 52A10, 52A38, 52A40, 49Q10.
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