Exercise 20, § 1

Let E be a real vector space of finite dimension, φ a bijective linear mapping of E onto its dual E*. Let A be a convex symmetric compact set in E, having 0 as an interior point; suppose that, for every point x of the boundary of A, the hyperplane of equation $\langle y - x, \varphi(x) \rangle = 0$ is a supporting hyperplane for A. a) Let $f(x) = |\langle x, \varphi(x) \rangle|$, and let a be a boundary point of A where $f(x)$ attains its minimum. Prove that, for every point b such that $\langle b, \varphi(a) \rangle = 0$, one also has $\langle a, \varphi(b) \rangle = 0$. (Remark that $\langle x, \varphi(x) \rangle \neq 0$ for $x \neq 0$, and that one may consequently suppose that $f(x) = \langle x, \varphi(x) \rangle \geqslant 0$; use the fact that every supporting hyperplane of A at the point a is also a supporting hyperplane of the set defined by $f(x) \leqslant f(a)$.) b) Prove that $(x, y) \mapsto \langle x, \varphi(y) \rangle$ is a symmetric bilinear form, and that A is identical with the set of points x such that $f(x) \leqslant \gamma$ for a suitable constant γ. (Reason by induction on the dimension of E, by considering, with the notation of a), the hyperplane of equation $\langle x, \varphi(a) \rangle = 0$.)