Exercise 20, § 2

a) In a vector space E, let A be a convex set that generates E and meets every straight line in a set that is closed relative to the straight line. Show that the following conditions are equivalent: α) There exists a line D such that D meets A in a compact segment that is not empty. β) There exists a line D such that every line parallel to D meets A in a compact segment. γ) A is distinct from E and is not a half-space determined by a hyperplane of E. (To show that $\gamma \Rightarrow \alpha$ use exerc. 14, d) of II, p. 67, and reduce to the case $E = \mathbf{R}^2$.) b) In a Hausdorff topological vector space E, let A be a closed convex set which contains an interior point. Show that if the frontier of A is a non-empty linear variety, then A is a closed half-space (use exerc. 14, d) of II, p. 67, to show that the frontier of A is necessarily a hyperplane, then apply a)).