Exercise 17, § 2

In a Hausdorff topological vector space E, a closed convex set A is said to be parabolic if, for every $z \notin A$, each half-line originating at z and contained in $z + C_A$ meets A. a) Give an example of a parabolic convex set A in $\mathbf{R}^2$ such that $C_A$ is not just a single half-line. b) Let A be a closed convex set in E such that $C_A \neq \{0\}$, but such that A is not parabolic. Show that if $z \notin A$ is such that $z + C_A$ contains a half-line D with end point z which does not meet A, then neither the convex envelope of $A \cup \{z\}$ nor the pointed cone with vertex z generated by A is closed in E. Further if D' is the closed half-line originating at z and opposite to D (so that $D = 2z - D'$) then $D' + A$ is not closed in E, and there exists a plane P containing D and a closed convex set $B \subset P$, such that $B \cap A = \varnothing$ but that the distance of B from $P \cap A$ (in any norm on P) is zero. c) In E, let A be a closed convex set that is locally compact and parabolic; show that if $B \subset E$, is closed and convex then $A - B$ is closed (same method as exerc. 16). d) Let A, A' be two closed convex subsets of E that are locally compact and parabolic; show that the convex envelope of $A \cup A'$ is closed in E (same method as in exerc. 16). Give an example in $\mathbf{R}^2$ where A is parabolic, $A'$ is non-parabolic and the convex envelope of $A \cup A'$ is not closed in $\mathbf{R}^2$.

e) In E, let A be a closed convex set that is locally compact and parabolic; show that for all $z \notin A$, the pointed convex cone with vertex z, generated by A, is closed in E (use d)).

f) Let $E_1, E_2$ be two Hausdorff topological spaces, $A_1$ (resp. $A_2$) a closed convex set in $E_1$ (resp. $E_2$) that is also parabolic. Show that the set $A_1 \times A_2$ is parabolic in $E_1 \times E_2$.

\* g) Show that a barrelled space of infinite dimension does not contain a parabolic closed convex set that is both locally compact and not compact.

h) Let $E_0 = l^2(\mathbf{N})$, and in $E_0$, let K be the set of points $x = (\xi_n)$ such that $|\xi_n| \leq 1/(n+1)$ for all n; K is compact. The cone E of vertex 0 generated by K is a vector subspace of $E_0$ and K is absorbent in E. Let p be the gauge of K in E; it is a lower semi-continuous function. In the normed product space $E \times \mathbf{R}$, show that the set A of points $(x, \zeta)$ such that $\zeta \geq (p(x))^2$ is closed, convex, parabolic and locally compact. Show that $A + (-A)$ and the convex envelope of $A \cup (-A)$ are not locally compact. \*