LOCAL LIPSCHITZ CONTINUITY OF THE DIAMETRIC COMPLETION MAPPING

Abstract. The diametric completion mapping associates with every closed bounded set C in a normed linear space the set γ(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 γ if the space is strictly convex, (ii) γ(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. The existence of nontrivial curves of constant width was already known to Eu-ler [8]. They have since become an active subject of study, appearing in many different contexts and having surprising applications in different areas of mathe-matics and engineering. The notion of diametrically maximal sets appeared later, at the beginning of last century, as a generalization of constant width sets [13]. A set in a metric space is diametrically maximal if the addition of any point to the set increases the diameter. Other names such as complete (which will also be used in this note), diametrically complete or entire sets have been used in the literature to refer to this concept. In this paper, we study diametrically maxima