LOCAL LIPSCHITZ CONTINUITY OF THE DIAMETRIC COMPLETION MAPPING

The diametric completion mapping associates with every closed bounded set C in a normed linear space the set gamma(C) of its completions, that is, of the diametrically complete sets containing C and having the same diameter. We prove local Lipschitz continuity of this set-valued mapping, with respect to two possible arguments: either as a function on the space of closed, bounded and convex sets, while the norm is fixed, or as a function on the space of equivalent norms, while the set C is fixed. In the first case, our result is valid in spaces with Jung constant less than 2, whereas the result in the second case is only proved for finite dimensional spaces. In this setting, we further show: (i) the maximal volume completion is a continuous selection for gamma if the space is strictly convex, (ii) gamma(C) is convex for all C if and only if the space has the property, studied by Eggleston, that every diametrically maximal set is of constant width.