Exercise 3, § 1

Let E be a real finite dimensional vector space with dimension $n$, $f$ a positive and separating symmetric bilinear form on E, and $B_f$ the bounded convex set defined by the relation $f(x, x) \leq 1$. If $a = (a_1, ..., a_n)$ is a basis of E and $\Delta$ the discriminant of $f$ with respect to this basis, we call the volume of $B_f$ with respect to $a$ the number $v_a(f) = \gamma_n |\Delta|^{-1/2}$, where $\gamma_n = \pi^{n/2}/\Gamma\left(\frac{n}{2} + 1\right)$. If $b = (b_1, ..., b_n)$ is a second basis of E, and if

we have $v_b(f) = |\delta| v_a(f)$. a) Show that, if $f$ and $g$ are two positive, separating symmetric bilinear forms such that $B_f \subset B_g$ (which is equivalent to $g \leq f$), then $v_a(f) \leq v_a(g)$ (consider a basis for E which is orthogonal for both $f$ and for $g$). b) Let A be a symmetric compact convex set in E, with 0 as an interior point. Show that among all positive, separating symmetric bilinear forms $f$ on E such that $A \subset B_f$, there exists one and only one for which the volume of $B_f$ (with respect to the given basis of E) is the smallest possible. (To show uniqueness observe that if A is contained in $B_f$ and $B_g$, it is in $B_{(f+g)/2}$ and that we have $v_a((f+g)/2) \leq \frac{1}{2}(v_a(f) + v_a(g))$ for every basis $a$ of E which is orthogonal for both $f$ and $g$.) c) Let A be a symmetric compact convex set in E, with 0 as an interior point, and let $f$ be the positive, separating, symmetric bilinear form such that $A \subset B_f$ and that $B_f$ has the smallest possible volume with respect to a given basis of E. Show that there exist points $x_1, ..., x_n, u_1, ..., u_n$ in E with the following properties : α) For every $k$, we have $x_k \in A$ and $f(x_k, x_k) = 1$. β) The sequence $(u_1, ..., u_n)$ is an orthonormal basis of E for $f$. γ) If we put $x_k = \sum_{j=1}^n a_{kj} u_j$ for $1 \leq k \leq n$, we have $a_{kj} = 0$ for $k < j$ and $a_{kk}^2 \geq (n - k + 1)/n$. (Argue by induction on $k$. Suppose $x_1, ..., x_k, u_1, ..., u_k$ have been constructed, let $P_k$ be the orthoprojector (for $f$) from E onto the subspace generated by $u_1, ..., u_k$; for every $\varepsilon > 0$, consider the bilinear form $f_\varepsilon$ defined by and prove that $A \not\subset B_{f_\varepsilon}$, using b). For every integer $p \geq 1$ choose a point $y_p$ in A not belonging to $B_{f_{1/p}}$; take for $x_{k+1}$ a limit point of the sequence $(y_p)$ such that next choose $u_{k+1}$.) d) Prove the analogues of b) and c) for the positive separating symmetric bilinear forms such that $B_f \subset A$ and for which the volume of $B_f$ (with respect to a given bases of E) is the largest possible.