Mazur’s intersection property and a Kreĭn-Mil′man type theorem for almost all closed, convex and bounded subsets of a Banach space

Abstract:

Let $\mathcal {V}$ (resp. ${\mathcal {V}^*}$) be the set of all closed, convex and bounded (resp. ${w^*}$-compact and convex) subsets of a Banach space $E$ (resp. of its dual ${E^*}$) furnished with the Hausdorff metric. It is shown that if there exists an equivalent norm $|| \cdot ||$ in $E$ with dual $|| \cdot |{|^*}$ such that $\left ( {E,|| \cdot ||} \right )$ has Mazur’s intersection property and $\left ( {{E^*},||\cdot |{|^*}} \right )$ has ${w^*}$-Mazur’s intersection property, then (1) there exists a dense ${G_\delta }$ subset ${\mathcal {V}_0}$ of $\mathcal {V}$ such that for every $X \in {\mathcal {V}_0}$ the strongly exposing functionals form a dense ${G_\delta }$ subset of ${E^*}$; (2) there exists a dense ${G_\delta }$ subset $\mathcal {V}_0^*$ of ${\mathcal {V}^*}$ such that for every ${X^*} \in \mathcal {V}_0^*$ the ${w^*}$-strongly exposing functionals form a dense ${G_\delta }$ subset of $E$. In particular every $X \in {\mathcal {V}_0}$ is the closed convex hull of its strongly exposed points and every ${X^*} \in \mathcal {V}_0^*$ is the ${w^*}$-closed convex hull of its ${w^*}$-strongly exposed points.