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On the strongly extreme points of convex bodies in separable Banach spaces. (English) Zbl 0758.46015

Let \(C\) be a bounded closed convex set in a Banach space \(X\). A point \(x\) in \(C\) is called a (weakly) strongly extreme point of \(C\) if for any sequences \((y_ n)\), \((z_ n)\) in \(C\) \(\lim_ n\| x-(y_ n+ z_ n)/2\|=0\) implies \(\lim\| y_ n-z_ n\|=0\) (weak-\(\lim(y_ n+z_ n)=0)\). The point \(x\) is called a weak\(^*\)-extreme point of \(C\) if it is an extreme point of the weak\(^*\)-closure of \(C\) in \(X^{**}\). The last two notions (weakly strongly extreme point and weak\(^*\)- extreme point) are equivalent.
Theorem 1. Let \(X\) be a separable Banach space. Then there exists an equivalent norm \(|\cdot|\) such that the unit ball of \((X,|\cdot|)\) has at most countably many strongly extreme points.
Theorem 2. Let \(X\) be a separable non-reflexive Banach space. Then \(X\) admits an equivalent norm such that the unit ball under the new norm has at most countably many weakly strongly extreme points, i.e. weak\(^*\)- extreme points.
Corollary. Let \(X\) be a separable Banach space. Then \(X\) is reflexive if and only if every bounded closed convex body in \(X\) has uncountably many weakly strongly extreme points, i.e. weak\(^*\)-extreme points.

MSC:

46B20 Geometry and structure of normed linear spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
46A55 Convex sets in topological linear spaces; Choquet theory
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