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 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 Euler [7]. They have since become an active subject of study, appearing in many different contexts and having surprising applications in different areas of mathematics and engineering. The notion of diametrically maximal sets appeared later, at the beginning of last century, as a generalization of constant width sets [12]. 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 maximal sets in normed linear spaces. It is well known that every bounded set C can be completed, in the sense that there is always a complete set D containing C such that diamC = diamD (see, for example, the survey [5]). Such a set D is called a completion of C. In general, a set can have infinitely many completions. We denote by H the family of all nonempty, closed, bounded and convex sets in our space, endowed with the Hausdorff metric. The diametric completion mapping γ : H → 2H associates with each C ∈ H the family of all its completions. The question with which we are mainly concerned in this paper can be phrased in a simple manner: is γ a continuous function? Every complete set satisfies the spherical intersection property and therefore it is an intersection of closed balls [5]. For this reason, γ is somehow related to the ball hull mapping β, which associates with every bounded set the intersection of all balls containing it. It is known that β need not be continuous, even in three dimensional spaces [15]. Hence, having in mind that γ = γ ◦ β, one might expect a similar behavior of γ. To our surprise, this is not so and we show that γ is locally Lipschitz continuous in the wide class of normed spaces with Jung constant less than 2. This class includes the spaces with 2010 Mathematics Subject Classification: 46B20, 52A05, 52A21 The first author was partially supported by the DGICYT project MTM 2009-07848.