Exercise 3, § 3

Let E be a vector space over $\mathbf{R}$ and let p be a sublinear function on E (II, p. 20). Let A be a convex set such that $\inf_{y \in A} p(y) > -\infty$. a) Show that the function

is a sublinear function on E such that $-p(-x) \leq q(x) \leq p(x)$. b) Show that there exists a linear form h on E such that $h(x) \leq p(x)$ in E and that $\inf_{y \in A} p(y) = \inf_{y \in A} h(y)$ (take h such that $h(x) \leq q(x)$).