Optimal sets for a class of minimization problems with convex constraints

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.