Strongly extreme points in $L^p(u,X)

Abstract

A natural characterization of strongly extreme points in the unit ball of L p (fi, X), where 1 < p < oo, is given. This characterization is compared to similar known results concerning strongly exposed points and extreme points. Sundaresan [8] and Johnson [6] considered the problem of characterizing extreme points in the unit ball of LP(/u, X), where X is a Banach space, (S, 2, ju) is a measure space and 1 < p < oo. It is easily shown that sufficient conditions for/to be such a point are that ||/||^=1 and, for almost all s in the support of/, the element f(s)/\\f(s)\\ is an extreme point of the unit ball of X. However, Greim [4] has produced an example of a nonseparable X and a norm one/in LP(X, X), where X is Lebesgue measure on [0,1], such that/is an extreme point of the unit ball of LP(X, X) but, fpr all s in [0, 1], the element f(s)/\\f(s)\\ is not an extreme point of the unit ball of X.