Optimal sets for a class of minimization problems with convex constraints

Chiara Bianchini (1) , Antoine Henrot (1)

1 CORIDA - Robust control of infinite dimensional systems and applications

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.

Mots clés: area; length; shape optimization; isoperimetric inequalities; Convex geometry

Domaines: Equations aux dérivées partielles [math.AP]; Mathématiques générales [math.GM]

Chiara Bianchini, Antoine Henrot. Optimal sets for a class of minimization problems with convex constraints. Journal of Convex Analysis, 2012, 19 (3), pp.725-758. ⟨hal-00549131⟩