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.
The radius of a compact set in a Hilbert space
AbstractIn this note we improve the formular(A)⩽
$$\frac{1}{{\sqrt 2 }}$$
δ(A) proved by Routledge for Hilbert spaces. We show that if A is a relatively compact set, thenr(A)<
$$\frac{1}{{\sqrt 2 }}$$
δ(A)
Asymptotic normal structure and fixed points of nonexpansive mappings
A mapping / defined on a subset X of a Banach space E and taking values in E is said to be nonexpansive if \flx) — fiy)\ < \x — y\ for all x,y e X. In this paper we introduce a promising new geometric property of Banach spaces and show that it yields via a minor modification of known arguments a new fixed point theorem for nonexpansive mappings which includes Kirk's famous result as well as a recent result of Karlovitz. We also discuss in detail a situation not covered by this result. 1. Statement of results. Throughout this paper, E denotes a Banach space with norm | |. If X is a subset of E and /: X —» E, then / is said to be nonexpansive if |/(x) — /(y)| < |x — y| for all x,y E X. The space E is said to have the BrowderGöhde-Kirk property (short: B-G-K property), if any nonexpansive self-mapping of a nonempty, closed, bounded and convex subset X of E has a fixed point in X. In 1965, F. E. Browder [2] and D. Göhde [6] independently proved that every uniformly convex Banach space has the B-G-K property, while W. A. Kirk [8] established this property for the wider class of reflexive Banach spaces having normal structure. (Recall that E is said to have normal structure, if for each bounded, closed and convex subset X of E consisting of more than one point there is a point x G A' such that sup{|y — x\: y E X) < diani| |(A") (= sup{|w — o|: u, v E X}).) Motivated by a certain method often employed in proofs of fixed point theorems for nonexpansive mappings (see Lemma 1 below), we introduce the following weakening of the concept of normal structure: A Banach space E is said to have asymptotic normal structure, if for each bounded, closed and convex subset X of E consisting of more than one point and each sequence {x„} in X satisfying x„ — xn+x -» 0 as n -» oo, there is a point x G X such that lim inf„_00|x„ — x| < diani| ¡(X). Theorem 1. Every reflexive Banach space with asymptotic normal structure has the B-G-K property. The interest in Theorem 1 stems, of course, from the fact that normal structure defines a narrower class of Banach spaces than asymptotic normal structure. This can be seen, among others, from our next result. Received by the editors July 31, 1979 and, in revised form, April 15, 1980. 1980 Mathematics Subject Classification. Primary 47H10.