A Characterization of Extremal Sets in Hilbert Spaces
We give a characterization of extremal sets in Hilbert spaces that generalizes a classical theorem of H. W. E. Jung. We investigate also the behaviour of points near to the circumsphere of such a set with respect to the Kuratowski and Hausdorff measures of non-compactness.
Perturbation of Sets and Centers
Given a bounded set A, a Chebyshev center (when it exists) is–in some sense–a candidate to give a global information on the set. Finding the centers of A is of great importance for applications. In many cases, it is very important to understand how they change when the set A is perturbed. Our main result is a new characterization of Hilbert spaces: in fact, we will show that the best estimate we can give in these spaces, concerning perturbations of sets, cannot be expected outside this class of spaces. Moreover, we collect, we partly sharpen and we reprove in a simple way most known results.
Normed Linear Spaces that are Uniformly Convex in Every Direction
The concept of uniform convexity in a normed linear space is based on the geometric condition that if two members of the unit ball are far apart, then their midpoint is well inside the unit ball. We consider here a generalization of this concept whose geometric significance is that the collection of all chords of the unit ball that are parallel to a fixed direction and whose lengths are bounded below by a positive number has the property that the midpoints of the chords lie uniformly deep inside the unit ball. This notion, called uniform convexity in every direction (UCED), was first used by A. L. Garkavi [5; 6] to characterize normed linear spaces for which every bounded subset has at most one Čebyŝev center. We discuss questions of renorming spaces so as to be UCED and forming products of spaces that are uniformly convex in every direction.