Exercise 6, § 7

Let E, E' be two vector spaces in separating duality, and A be a convex set in E containing 0 and closed for $\sigma(E, E')$. For all $a \in A$, the set $F'_a$ of points $x' \in A^\circ$ such that $\langle a, x' \rangle = -1$ is a closed (for $\sigma(E', E)$), convex set of $A^\circ$. Show that $F'_a$ is the facet in $A^\circ$ of each of the internal points of $F'_a$ relative to the linear variety generated by $F'_a$. We say that $F'_a$ is the dual facet of $a$ in $A^\circ$. If $F_a$ is the facet of $a$ in A, show that $F'_a$ is also the dual facet in $A^\circ$ of each of the internal points of $F_a$ relative to the linear variety generated by $F_a$; further, if A is identified with $A^{\circ\circ}$, the dual facet in $A$ of each internal point of $F'_a$ relative to the linear variety generated by $F'_a$, contains $F_a$. When $F'_a$ is not empty (which is always the case when $E$ is finite-dimensional and $a \neq 0$, cf. II, p. 78, exerc. 13), we say that each of $F_a, F'_a$ is the dual facet of the other.

We say that a point $a \in A$ is smooth point of $A$ if $F'_a$ is a single point (in other words if there exists one closed hyperplane of support of $A$ passing through $a$); we say that $a$ is a point of strict convexity (or is an exposed point) if there exists a closed hyperplane $H$ supporting $A$ so that $H \cap A = \{a\}$; this is the same as saying that there exists an internal point of $F'_a$ (relative to a linear variety generated by $F'_a$) which is a smooth point of $A^\circ$.