Exercise 4, § 3

Let A be a non-empty set of E, a vector space over $\mathbf{R}$, and p a sub-linear function on E. Let B be the set of $z \in E$ such that $\inf_{x \in A} p(x - z) \leq 0$; we have $A \subset B$ and $\inf_{x \in A} p(x) \leq p(z)$ for all $z \in B$; from which $\inf_{x \in A} p(x) = \inf_{z \in B} p(z)$. a) Show that the set of the $y \in E$ such that $\inf_{z \in B} p(z - y) \leq 0$ is the set B. b) Deduce from a) that the intersection of B and any affine line D in E is closed in D (show that, whatever the points a, b of E, the function $t \to p(a + tb)$ is continuous in $\mathbf{R}$). c) Suppose that for each pair of points x, y of A there exists $z \in A$ such that $p(z - \frac{1}{2}(x + y)) \leq 0$. Show that, for each pair of points u, v of B we have $\frac{1}{2}(u + v) \in B$ (write

for $x, y, z$ in A). Deduce that B is then convex (use b)).

d) Under the hypotheses of c), show that there exists a linear form h on E such that $h(x) \leq p(x)$ and that we have $\inf_{y \in A} p(y) = \inf_{y \in A} h(y)$ (use c) and exerc. 3).