Exercise 20, § 1

Let $E$ be a real finite dimensional vector space, and $\phi$ be a bijective linear mapping from $E$ onto its dual $E^*$. Let $A$ be a symmetric compact convex set in $E$, with 0 as an interior point; assume that for every $x$ in the boundary of $A$, the hyperplane with equation $\langle y - x, \phi(x) \rangle = 0$ is a support hyperplane for $A$. a) Let $f(x) = |\langle x, \phi(x) \rangle|$, and let $a$ be a boundary point of $A$ where $f(x)$ attains its minimum. Show that for every point $b$ such that $\langle b, \phi(a) \rangle = 0$, we also have $\langle a, \phi(b) \rangle = 0$. (Observe that $\langle x, \phi(x) \rangle \neq 0$ for $x \neq 0$, and so we can assume that $f(x) = \langle x, \phi(x) \rangle \geq 0$; use the fact that every support hyperplane of $A$ at the point $a$ is also a support hyperplane of the set defined by $f(x) \leq f(a)$.) b) Show that $(x, y) \mapsto \langle x, \phi(y) \rangle$ is a symmetric bilinear form, and that $A$ is identical with the set of all points $x$ such that $f(x) \leq \gamma$ for a suitable constant $\gamma$. (Argue by induction on the dimension of $E$; with the notations of $a$, consider the hyperplane with the equation $\langle x, \phi(a) \rangle = 0$.)