Isometries of the class of ball-bodies

We characterize the surjective isometries, with respect to the Hausdorff distance, of the class of bodies given by intersections of Euclidean unit balls. We show that any such isometry is given by the composition of a rigid motion with either the identity or the $c$-duality mapping, which maps a body in this class to the intersection of Euclidean unit balls centered at points in the body.