Inserted: 23 apr 2020
Last Updated: 8 feb 2021
Journal: Journal de l'École polytechnique - Mathématiques
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 Bonnesen, Fenchel: Theorie der konvexen Körper,Springer-Verlag, 1934. 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.