Exercise 32, § 7

Let $E$ and $F$ be two Hausdorff locally convex spaces, let $A$ be a convex subset of $E$, and $u$ a linear mapping of $E$ in $F$. a) The inverse image under $u$ of a support variety of $u(A)$ (II, p. 87, exerc. 3, c)) is a support variety of $A$. b) If $A$ is compact and $u$ is continuous, then every extremal point of $u(A)$ is the image under $u$ of an extremal point of $A$. c) If $A$ is a locally compact cone with vertex 0 and if $u$ is continuous then every extremal generator of $u(A)$ is the image under $u$ of an extremal generator of $A$.