Strongly extreme points and the Radon-Nikodým property

Abstract:

We prove that if $K$ is a bounded and convex subset of a Banach space $X$ and $x$ is a point in $K$, then $x$ is a strongly extreme point of $K$ if and only if $x$ is a strongly extreme point of ${\overline K ^{\ast }}$ which is the weak$^{{\ast }}$ closure of $K$ in ${X^{{\ast }{\ast }}}$. We also prove that a Banach space $X$ has the Radon-Nikodým property if and only if for any equivalent norm on $X$, the unit ball has a strongly extreme point.