Exercise 21, § 7

a) In a Hausdorff locally convex space E, let A be a locally compact closed convex cone, that does not contain any line. Show that A is a cone of compact sole (apply prop. 2 of II, p. 55, to the vertex of A, which is an extremal point of A). Deduce that there exists a closed support hyperplane H of A, which contains the vertex s of A and is such that $H \cap A = \{ s \}$. b) Let A, B be two closed convex cones with vertex 0 in E, that are locally compact and do not contain a line. Show that if $A \cap B = \{ 0 \}$, then $A - B$ is a closed, locally compact, cone not containing any line (method as in II, p. 67, exerc. 16). Deduce that there exists a closed hyperplane that supports both A and B, that separates A from B and such that $H \cap A = H \cap B = \{ 0 \}$. c) Give an example of a locally compact closed convex cone A such that $A - A$ is not locally compact (cf. II, p. 78, exerc. 11). d) Let A be a locally compact closed convex set in E, which does not contain a line. Let $x_0$ be a point of A, $C_A$ the asymptotic cone of A (cf. II, p. 67, exerc. 14) and H a support hyperplane of $x_0 + C_A$ passing through $x_0$ and such that $(x_0 + C_A) \cap H = \{ x_0 \}$. If $f(x) = a$ is the equation of H and if $f(x) \geq a$ in $x_0 + C_A$, then show that for every real number b, the set of the $y \in A$ such that $f(y) \leq b$ is compact.