In this paper we address the following shape optimization problem: find the
planar domain of least area, among the sets with prescribed constant width and
inradius. In the literature, the problem is ascribed to Bonnesen, who proposed
it in \cite{BF}. In the present work, we give a complete answer to the problem,
providing an explicit characterization of optimal sets for every choice of
width and inradius. These optimal sets are particular Reuleaux polygons.Comment: to appear in Journal Ecole Polytechniqu