A full classification of the isometries of the class of ball‐bodies

Complementing our previous results, we give a classification of all isometries (not necessarily surjective) of the metric space consisting of ball‐bodies, endowed with the Hausdorff metric. ‘Ball‐bodies’ are convex bodies which are intersections of translates of the Euclidean unit ball. We show that any such isometry is either a rigid motion or a rigid motion composed of the ‐duality mapping. In particular, any isometry on this metric space has to be surjective.