On drop property for convex sets

Let (X, ‖ · ‖) be a real Banach space. Let C be a closed convex set in X. By a drop D(x, C) determined by a point x ∈ X, x / ∈ C, we shall mean the convex hull of the set {x} ∪ C. We say that C has the drop property if C 6= X and if for every nonvoid closed set A disjoint with C, there exists a point a ∈ A such that D(a, C) ∩ A = {a}. For a given C a sequence {xn} in X will be called a stream if xn+1 ∈ D(xn, C) \ C (cf. [6]). When the set A has a positive distance from C, a variety of “Drop theorems” has been obtained in [1, 2, 3] and [8]. If C is the closed unit ball and has the drop property then we say that the norm ‖ · ‖ has the drop property [9]. Norms with the drop property have been investigated in papers [4, 6] and [9]. The drop property for closed bounded sets has been considered in [5]. There was proved that a bounded closed convex symmetric set having the drop property is compact or has a nonempty interior. We shall prove this theorem without assumptions on boundness and symmetry of sets under consideration. The Kuratowski measure of noncompactness of a set G in a Banach space X is the infimum α(G) of those ε > 0 for which there is a covering of G by a finite number of sets of diameter less than ε. For a closed convex set C denote by F (C) the set of all linear continuous functionals f ∈ X, f 6= 0, which are bounded above on C. For f ∈ F (C) and δ > 0 put