Exercise 15, § 2

a) Let A be a closed convex subset of a Hausdorff topological vector space E. If for some point $x_0 \in A$ there exists a neighbourhood V of $x_0$ in E such that $V \cap A$ is compact, then show that A is locally compact. Deduce that the closure in E of a locally compact convex set is locally compact. b) Let A be a closed convex set that is locally compact but not compact in E; show that the asymptotic cone (exerc. 14) is not the single point $\{0\}$.