Optimal sets for a class of minimization problems with convex constraints
Résumé
We look for the minimizers of the functional $J_\lambda(\Omega) = \lambda|\Omega|-P(\Omega)$ among planar convex domains constrained to lie into a given ring. We prove that, according to the values of the parameter $\lambda$, 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.
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