For all $n\in\mathbb{N}$, define $$\Delta(n) = \left\{F\left(k\ln(2), \frac{2\pi l}{2^k}\right),\, k\in\{0,\ldots,n\},\, l\in\{1,\ldots,2^k\}\right\}.$$ Fix $r>0$ as in question 6.6. Show that there exists a constant $A\geq 1$ satisfying the following two properties:

1. for all $g\in\Gamma(n\ln(2))$, $$|\{v\in\Delta(n) \text{ such that } d(gv_0,v)\leq r\}| \leq A,$$
2. for all $v\in\Delta(n)$, $$|\{g\in\Gamma(n\ln(2)) \text{ such that } d(gv_0,v)\leq r\}| \leq A.$$

Show the existence of constants $C_1 > C_2 > 0$ and $R_0 > 0$ such that, for all $R\geq R_0$, $$C_2 e^R \leq |\Gamma(R)| \leq C_1 e^R.$$

Deduce the existence of constants $C_1' > C_2' > 0$ and $s_0 > 1$ such that, for all $k\in\mathbb{N}^*$ and all $s\geq s_0$, $$C_2' s\,|P_k\cap T| \leq |P_k(s)| \leq C_1' s\,|P_k\cap T|.$$

For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \operatorname{det}(\operatorname{Gram}(e, u)).$$
(a) Let $M \in \mathcal{M}_p(\mathbb{R})$, $e = (e_1, \ldots, e_p)$ and $e^{\prime} = (e_1^{\prime}, \ldots, e_p^{\prime})$ in $E^p$ satisfying $e_i^{\prime} = \sum_{j=1}^p M_{ij} e_j$ for all $i \in \llbracket 1, p \rrbracket$. Show that $\Omega_p(e^{\prime}) = \operatorname{det}(M) \Omega_p(e)$.
(b) Let $e \in E^p$. Show that $\Omega_p(e) \neq 0$ if and only if $e$ is a free family.
(c) Verify that $\Omega_p(e)(e) \geqslant 0$ for all families $e \in E^p$.

Let $C$ be a non-empty closed convex subset of $\mathbb{R}^d$ and let $\sigma_C : \mathbb{R}^d \rightarrow \mathbb{R} \cup \{+\infty\}$ be defined by: $$\sigma_C(p) := \sup\{p \cdot x, x \in C\}$$ show that $$C = \left\{x \in \mathbb{R}^d : p \cdot x \leqslant \sigma_C(p), \forall p \in \mathbb{R}^d\right\}$$ (so that $C$ is an intersection of closed half-spaces).

Let $C$ be a non-empty closed convex subset of $\mathbb{R}^d$ and let $\sigma_C : \mathbb{R}^d \rightarrow \mathbb{R} \cup \{+\infty\}$ be defined by: $$\sigma_C(p) := \sup\{p \cdot x, x \in C\}$$ show that $$C = \left\{x \in \mathbb{R}^d : p \cdot x \leqslant \sigma_C(p), \forall p \in \mathbb{R}^d\right\}$$ (so that $C$ is an intersection of closed half-spaces).

For $\ell \in \mathcal { L } ( E , \mathbb { R } )$, we denote by $\left. \ell \right| _ { F }$ the restriction of $\ell$ to $F$. Show that the restriction map $r _ { F } : \begin{array} { c c c } \mathcal { L } ( E , \mathbb { R } ) & \rightarrow & \mathcal { L } ( F , \mathbb { R } ) \\ \ell & \mapsto & \left. \ell \right| _ { F } \end{array}$ is surjective.

Show that there exist $K$ polynomials $L_{1}, \ldots, L_{K}$ in $\mathbb{R}_{K-1}[X]$ such that, for any function $f \in \mathcal{C}^{K}([0,1])$, the polynomial $P = \sum_{j=1}^{K} f\left(x_{j}\right) L_{j}$ satisfies $$\forall \ell \in \llbracket 1, K \rrbracket, \quad P\left(x_{\ell}\right) = f\left(x_{\ell}\right).$$

For all $e \in E^p$, we call the $p$-volume of $e$ the quantity $$\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = \left(\operatorname{det}(\operatorname{Gram}(e, e))\right)^{1/2}.$$
(a) Calculate $\operatorname{vol}_p(b)$ when $b = (b_1, \ldots, b_p)$ is an orthonormal family of vectors of $E$.
(b) Suppose here that $p \geqslant 2$. Let $e = (e_1, \ldots, e_p) \in E^p$. We denote by $\operatorname{pr}$ the orthogonal projection onto the orthogonal of the space spanned by the family $e_2^p = (e_2, \ldots, e_p)$. Show that $\operatorname{vol}_p(e) = \|\operatorname{pr}(e_1)\| \operatorname{vol}_{p-1}(e_2^p)$.
(c) For all free families $e = (e_1, \ldots, e_p) \in E^p$, show that $\operatorname{vol}_p(e) \leqslant \prod_{i=1}^p \|e_i\|$ with equality if and only if $e$ is a family of pairwise orthogonal vectors.

Let $A$ be a non-empty convex subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d \backslash A$, show that there exists $p \in \mathbb{R}^d \backslash \{0\}$ such that $$p \cdot x \leqslant p \cdot y, \forall y \in A$$

Let $A$ be a non-empty convex subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d \setminus A$, show that there exists $p \in \mathbb{R}^d \setminus \{0\}$ such that $$p \cdot x \leqslant p \cdot y, \forall y \in A.$$

For $\ell \in \mathcal { L } ( E , \mathbb { R } )$, we denote by $\left. \ell \right| _ { F }$ the restriction of $\ell$ to $F$. The restriction map is $r _ { F } : \mathcal { L } ( E , \mathbb { R } ) \rightarrow \mathcal { L } ( F , \mathbb { R } )$, $\ell \mapsto \left. \ell \right| _ { F }$, and $d_{\omega} : E \rightarrow \mathcal{L}(E,\mathbb{R})$, $x \mapsto \omega(x,\cdot)$. The $\omega$-orthogonal is $F ^ { \omega } = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$.
Specify the kernel of $r _ { F } \circ d _ { \omega }$. Deduce that $\operatorname { dim } F ^ { \omega } = \operatorname { dim } E - \operatorname { dim } F$.

We fix $f \in \mathcal{C}^{K}([0,1])$ and denote by $P$ the polynomial determined in question Q7. For all $k \in \llbracket 0, K-1 \rrbracket$, show that there exist at least $K - k$ distinct real numbers in $[0,1]$ at which the function $f^{(k)} - P^{(k)}$ vanishes.

For all $e \in E^p$, we call the $p$-volume of $e$ the quantity $$\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = \left(\operatorname{det}(\operatorname{Gram}(e, e))\right)^{1/2}.$$
(a) Show that if $e \in E^p$ is a free family and if $b \in E^p$ is an orthonormal basis of $\operatorname{Vect}(e)$, then $\operatorname{vol}_p(e) = \left|\operatorname{det}\left(P_b^e\right)\right|$ where $P_b^e$ is the change of basis matrix from $b$ to $e$, i.e. $e_j = \sum_{i=1}^p \left(P_b^e\right)_{ij} b_i$ for all $j \in \llbracket 1, p \rrbracket$.
(b) Show that for all $e, e^{\prime} \in E^p$, we have $\left|\Omega_p(e)(e^{\prime})\right| \leqslant \operatorname{vol}_p(e) \operatorname{vol}_p(e^{\prime})$.

Let $A$ be a non-empty convex subset of $\mathbb{R}^d$. Let $I \in \mathbb{N}^*, x_1, \ldots, x_I \in A^I$ and $(\lambda_1, \ldots, \lambda_I) \in \mathbb{R}_+^I$ such that $\sum_{i=1}^I \lambda_i = 1$, show that:

• a) $\sum_{i=1}^I \lambda_i x_i \in A$,
• b) if $x := \sum_{i=1}^I \lambda_i x_i \in \operatorname{Ext}(A)$ then $x_i = x$ for all $i \in \{1, \ldots, I\}$ such that $\lambda_i > 0$.

Let $A$ be a non-empty convex subset of $\mathbb{R}^d$. Let $I \in \mathbb{N}^*, x_1, \ldots, x_I \in A^I$ and $(\lambda_1, \ldots, \lambda_I) \in \mathbb{R}_+^I$ such that $\sum_{i=1}^I \lambda_i = 1$, show that:
a) $\sum_{i=1}^I \lambda_i x_i \in A$,
b) if $x := \sum_{i=1}^I \lambda_i x_i \in \operatorname{Ext}(A)$ then $x_i = x$ for all $i \in \{1, \ldots, I\}$ such that $\lambda_i > 0$.

Let $n \in \mathbb{N}^*$ and $$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$ Show that $T _ { n }$ is the unique polynomial with real coefficients satisfying the relation $$\forall \theta \in \mathbb { R } , \quad T _ { n } ( \cos ( \theta ) ) = \cos ( n \theta ).$$

We denote $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. Prove that $C$ is a closed subset of $\mathcal{M}_{n,1}(\mathbb{R})$.

Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\alpha = (i_1, \ldots, i_p) \in \mathcal{I}_p$, we denote $e_{\alpha} = (e_{i_1}, \ldots, e_{i_p}) \in E^p$ and for all $\omega$ and $\omega^{\prime}$ elements of $\mathscr{A}_p(E, \mathbb{R})$ $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha}).$$
(a) Show that for all $\omega \in \mathscr{A}_p(E, \mathbb{R})$, we have $\omega = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \Omega_p(e_{\alpha})$.
(b) Deduce that $(\omega, \omega^{\prime}) \mapsto \langle\omega, \omega^{\prime}\rangle$ is an inner product on $\mathscr{A}_p(E, \mathbb{R})$ for which $(\Omega_p(e_{\alpha}))_{\alpha \in \mathcal{I}_p}$ is an orthonormal basis of $\mathscr{A}_p(E, \mathbb{R})$ and give the dimension of $\mathscr{A}_p(E, \mathbb{R})$.
(c) Construct in the case $p = d-1$ an isometry between $\mathscr{A}_p(E, \mathbb{R})$ and $E$.

Let $E$ be a subset of $\mathbb{R}^d$. Recall that $$\operatorname{co}(E) := \left\{\sum_{i=1}^I \lambda_i x_i, I \in \mathbb{N}^*, \lambda_i \geq 0, \sum_{i=1}^I \lambda_i = 1, (x_1, \ldots, x_I) \in E^I\right\}.$$ Show that $\operatorname{co}(E)$ is the smallest convex set containing $E$ and that $\operatorname{Ext}(\operatorname{co}(E)) \subset E$.

Let $E$ be a subset of $\mathbb{R}^d$. Recall that $$\operatorname{co}(E) := \left\{\sum_{i=1}^I \lambda_i x_i, I \in \mathbb{N}^*, \lambda_i \geq 0, \sum_{i=1}^I \lambda_i = 1, (x_1, \ldots, x_I) \in E^I\right\}.$$ Show that $\operatorname{co}(E)$ is the smallest convex set containing $E$ and that $\operatorname{Ext}(\operatorname{co}(E)) \subset E$.

Show that there exists a constant $C > 0$ for which the interpolation inequality $$\forall f \in \mathcal{C}^{K}([0,1]), \quad \max_{0 \leqslant k \leqslant K-1} \left\|f^{(k)}\right\|_{\infty} \leqslant \left\|f^{(K)}\right\|_{\infty} + C \sum_{\ell=1}^{K} \left|f\left(x_{\ell}\right)\right|$$ is satisfied.

Let $n \in \mathbb{N}^*$ and $$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$ For $k \in \llbracket 1 , n \rrbracket$, we set $y _ { k , n } = \cos \left( \frac { ( 2 k - 1 ) \pi } { 2 n } \right)$. Show that $$T _ { n } ( X ) = 2 ^ { n - 1 } \prod _ { k = 1 } ^ { n } \left( X - y _ { k , n } \right).$$

We denote $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. Deduce that the application $U \mapsto \left| U^\top A U \right|$ admits a maximum on $C$.

Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\alpha = (i_1, \ldots, i_p) \in \mathcal{I}_p$, we denote $e_{\alpha} = (e_{i_1}, \ldots, e_{i_p}) \in E^p$ and for all $\omega$ and $\omega^{\prime}$ elements of $\mathscr{A}_p(E, \mathbb{R})$ $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha}).$$
We consider $u, v \in E^p$. Show that $$\Omega_p(u)(v) = \langle\Omega_p(u), \Omega_p(v)\rangle.$$