Let $n \geq 1$ be an integer and $P \subset \mathbb{R}^n$ a polytope. Show that $P$ has a finite number of faces and at least one vertex.

13. Let $\mathscr { B } = \left( a _ { 0 } , \ldots , a _ { n } \right)$ be the unique orthogonal basis of $\left( \mathbb { R } _ { n } [ X ] , \langle \cdot , \cdot \rangle \right)$ such that $a _ { i }$ is a monic polynomial of degree $i$ for all $0 \leqslant i \leqslant n$. Show that, for all $0 \leqslant j \leqslant n - 1$, the coefficients of the polynomial $\prod _ { \ell = j + 1 } ^ { n } \left( X - r _ { \ell } \right)$ in the basis $\mathscr { B }$ are strictly positive real numbers. Hint: one may denote $\left( q _ { j , 0 } , \ldots , q _ { j , n - j } \right)$ the basis of $\left( \mathbb { R } _ { n - j } [ X ] , \langle \cdot , \cdot \rangle _ { j } \right)$ obtained in questions 8a and 8b and reason by descending induction on $j$.

Third Part

Let $\lambda$ be a strictly positive real number. For all real $x$ and $r$ such that $| x | < 1$ and $| r | < 1$, we set
$$F _ { \lambda } ( x , r ) = \left( 1 - 2 r x + r ^ { 2 } \right) ^ { - \lambda }$$
Show that the function $F _ { \lambda }$ is of class $\mathscr { C } ^ { \infty }$ on $] - 1,1 \left[ ^ { 2 } \right.$.

Let $n \geq 1$ be an integer and $P \subset \mathbb{R}^n$ a polytope. Let $V$ be the set of vertices of $P$. Show that $P = \operatorname{Conv}(V)$.

Prove that the number of blocks of each type is determined by the data of the three dimensions $d_j = \dim V_j$ ($1 \leqslant j \leqslant 3$) and the three ranks $r_1 = \operatorname{rg} u_1$, $r_2 = \operatorname{rg} u_2$ and $r_{21} = \operatorname{rg}(u_2 \circ u_1)$.

Let $V \subset \mathbb{R}^n$ be a non-empty finite set. Justify that to prove that $\operatorname{Conv}(V)$ is a polytope it suffices to treat the case where $\operatorname{Conv}(V)$ is not contained in a hyperplane of $\mathbb{R}^n$ and contains 0 in its interior.

Let $(A, B)$ and $(A', B')$ be two pairs of matrices in $\mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$. Prove that the following conditions are equivalent: (i) $(A, B)$ and $(A', B')$ are simultaneously equivalent; (ii) there exist $P \in \mathrm{GL}_m(\mathbb{C})$ and $Q \in \mathrm{GL}_n(\mathbb{C})$ such that $A' = QAP^{-1}$ and $B' = PBQ^{-1}$; (iii) there exists $R \in \mathrm{GL}_{m+n}(\mathbb{C})$ such that $M_{A',B'} = RM_{A,B}R^{-1}$ and $H = RHR^{-1}$.

Calculate $d _ { 2 } , d _ { 3 }$ and $d _ { 4 }$, then show that $d _ { n } \leqslant n !$ for all natural integer $n \in \mathbb { N } ^ { * }$.

Let $V \subset \mathbb{R}^n$ be a non-empty finite set. We assume that $\operatorname{Conv}(V)$ is not contained in a hyperplane of $\mathbb{R}^n$ and contains 0 in its interior. Show that the set $Q$ defined by $$Q = \left\{\ell \in \mathbb{R}^n : \langle \ell, x \rangle \leq 1 \quad \forall x \in V\right\}$$ is a polytope of $\mathbb{R}^n$.

Let $V \subset \mathbb{R}^n$ be a non-empty finite set. We assume that $\operatorname{Conv}(V)$ is not contained in a hyperplane of $\mathbb{R}^n$ and contains 0 in its interior, and that the set $Q = \{\ell \in \mathbb{R}^n : \langle \ell, x \rangle \leq 1\ \forall x \in V\}$ is a polytope of $\mathbb{R}^n$. Deduce that $\operatorname{Conv}(V)$ is a polytope.

In this question, we assume that $M$ is nilpotent. Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are respectively of the form $$\left(\begin{array}{cc} 0_r & B_0 \\ A_0 & 0_s \end{array}\right) \quad \text{and} \quad \left(\begin{array}{cc} \mathrm{I}_r & 0 \\ 0 & -\mathrm{I}_s \end{array}\right),$$ where $r$ and $s$ are natural integers with $|r - s| \leqslant 1$ and $A_0$ and $B_0$ form one of the following pairs: $$A_0 = \left(\begin{array}{ccccc} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)_{s \times (s+1)} \quad \text{and} \quad B_0 = \left(\begin{array}{ccccc} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{array}\right)_{(s+1) \times s};$$ $$A_0 = \mathrm{I}_r \quad \text{and} \quad B_0 = J_r;$$ $$A_0 = J_r \quad \text{and} \quad B_0 = \mathrm{I}_r;$$ $$A_0 = \left(\begin{array}{cccc} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{array}\right)_{(r+1) \times r} \quad \text{and} \quad B_0 = \left(\begin{array}{ccccc} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)_{r \times (r+1)}.$$

We will admit that for every non-empty closed convex set $C \subset \mathbb{R}^n$ and every $x \in \mathbb{R}^n \backslash C$, there exists a unique $y \in C$ such that $\langle x - y, z - y \rangle \leq 0$ for all $z \in C$.
Let $V \subset \mathbb{R}^n$ be a non-empty finite set. Prove that every vertex of $\operatorname{Conv}(V)$ belongs to $V$.

In this question, we assume that $M$ is invertible. Prove that $m = n$ and that $A$ and $B$ are invertible.

Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are of even size and are respectively of the form $$\left(\begin{array}{cc} 0_r & B_1 \\ A_1 & 0_r \end{array}\right) \quad \text{and} \quad \left(\begin{array}{cc} \mathrm{I}_r & 0_r \\ 0_r & -\mathrm{I}_r \end{array}\right),$$ where $$A_1 = \mathrm{I}_r \quad \text{and} \quad B_1 = \lambda \mathrm{I}_r + J_r$$ for $r$ nonzero integer and $\lambda$ nonzero complex suitable.

Let $\mathcal{F}_n$ be the $\mathbb{R}$-vector space of functions $f : \mathbb{R}^n \rightarrow \mathbb{R}$. For all $X \subset \mathbb{R}^n$, we denote $\mathbb{1}_X$ the indicator function of $X$. Let $\mathcal{U}_n$ be the vector subspace of $\mathcal{F}_n$ generated by the functions $\mathbb{1}_P$ where $P$ is a polytope of $\mathbb{R}^n$.
Let $f \in \mathcal{U}_1$. Prove that for all $x \in \mathbb{R}$ the limit of $f(y)$ as $y$ tends to $x$ while satisfying $y > x$, denoted $\lim_{y \rightarrow x^+} f(y)$, exists and that there exist finitely many reals $x \in \mathbb{R}$ such that $f(x) \neq \lim_{y \rightarrow x^+} f(y)$.

19. Show that the polynomials $T _ { n }$ are functions of positive type in dimension 2 .
Hint: you may use the exponential form of the cosine. We shall admit, in the rest of the problem, that for every integer $n \geqslant 0$ and every integer $N \geqslant 4$, the polynomial $a _ { n } ^ { \left( \frac { N } { 2 } - 1 \right) }$ is of positive type in dimension $N$. For an integer $N \geqslant 2$, we say that a polynomial $P \in \mathbb { R } [ X ]$ is $N$-conductive if, for every absolutely monotone function $f$ from $[ - 1,1 ]$ to $\mathbb { R }$, the polynomial $H ( f , P )$ is a function of positive type in dimension $N$.

Let $\mathcal{F}_n$ be the $\mathbb{R}$-vector space of functions $f : \mathbb{R}^n \rightarrow \mathbb{R}$. For all $X \subset \mathbb{R}^n$, we denote $\mathbb{1}_X$ the indicator function of $X$. Let $\mathcal{U}_n$ be the vector subspace of $\mathcal{F}_n$ generated by the functions $\mathbb{1}_P$ where $P$ is a polytope of $\mathbb{R}^n$.
We assume that $n > 1$. Let $f \in \mathcal{U}_n$. For $z \in \mathbb{R}$, we define the function $f_z : \mathbb{R}^{n-1} \rightarrow \mathbb{R}$ by $f_z(x_1, \ldots, x_{n-1}) = f(x_1, \ldots, x_{n-1}, z)$ for all $(x_1, \ldots, x_{n-1}) \in \mathbb{R}^{n-1}$. Prove that $f_z \in \mathcal{U}_{n-1}$.

20. Let $P _ { 1 }$ and $P _ { 2 }$ be two $N$-conductive polynomials. Show that if $P _ { 1 }$ is of positive type in dimension $N$, then $P _ { 1 } P _ { 2 }$ is $N$-conductive. We fix an integer $N \geqslant 4$ and an integer $n \geqslant 2$. We admit that the polynomial $a _ { n } ^ { \left( \frac { N } { 2 } - 1 \right) }$ has $n$ simple real roots $r _ { 1 } > r _ { 2 } > \cdots > r _ { n }$ in $] - 1,1 [$. Let $f : [ - 1,1 ] \rightarrow \mathbb { R }$ be an absolutely monotone function.

Let $\beta = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { 10 ^ { n ! } }$.
Justify that $\beta$ is well-defined, then show that $\beta$ is an irrational number.

Let $\mathcal{F}_n$ be the $\mathbb{R}$-vector space of functions $f : \mathbb{R}^n \rightarrow \mathbb{R}$. For all $X \subset \mathbb{R}^n$, we denote $\mathbb{1}_X$ the indicator function of $X$. Let $\mathcal{U}_n$ be the vector subspace of $\mathcal{F}_n$ generated by the functions $\mathbb{1}_P$ where $P$ is a polytope of $\mathbb{R}^n$.
Prove that the following definition allows us to define a linear form $\chi_n : \mathcal{U}_n \rightarrow \mathbb{R}$. We define $\chi_1(f)$ by the sum $\chi_1(f) = \sum_{x \in \mathbb{R}} \left(f(x) - \lim_{y \rightarrow x^+} f(y)\right)$, then for $f \in \mathcal{U}_n$ with $n > 1$, we set $$\chi_n(f) = \chi_1(g) \text{ with } g \text{ defined by } g(z) = \chi_{n-1}(f_z) \text{ for } z \in \mathbb{R}.$$ We will show at the same time the formula $\chi_n(\mathbb{1}_P) = 1$ for every polytope $P$ of $\mathbb{R}^n$ and we will justify that $\chi_n$ is independent of the coordinate system, namely that for every invertible linear map $A : \mathbb{R}^n \rightarrow \mathbb{R}^n$ we have $\chi_n(f \circ A) = \chi_n(f)$ for all $f \in \mathcal{U}_n$.

21. Show that the polynomial $H \left( f , \prod _ { i = 1 } ^ { n } \left( X - r _ { i } \right) \right)$ is a function of positive type in dimension $N$.

Let $\mathcal{U}_n$ be the vector subspace of functions $\mathbb{R}^n \rightarrow \mathbb{R}$ generated by indicator functions of polytopes of $\mathbb{R}^n$. For a polytope $P$ of $\mathbb{R}^n$, the relative interior $P^\circ$ is defined as $P^\circ = \{x \in P : \ell_i(x) = a_i \Leftrightarrow i \in S_F\}$ where $S_F = \{i \in I, \ell_i(x) = a_i\ \forall x \in F\}$.
Show that for every polytope $P$ of $\mathbb{R}^n$ and for all $x \in P \backslash P^\circ$, there exists a face $F \subset P$ such that $F \neq P$ and $x \in F$.

Let $\mathcal{U}_n$ be the vector subspace of functions $\mathbb{R}^n \rightarrow \mathbb{R}$ generated by indicator functions of polytopes of $\mathbb{R}^n$, and $\chi_n : \mathcal{U}_n \rightarrow \mathbb{R}$ the linear form defined recursively. For a polytope $P$ of $\mathbb{R}^n$, let $P^\circ$ denote its relative interior.
Show that for every polytope $P$ of $\mathbb{R}^n$, $\mathbb{1}_{P^\circ} \in \mathcal{U}_n$ and $\chi_n(\mathbb{1}_{P^\circ}) = (-1)^{\operatorname{dim} P}$.

Let $\mathcal{U}_n$ be the vector subspace of functions $\mathbb{R}^n \rightarrow \mathbb{R}$ generated by indicator functions of polytopes of $\mathbb{R}^n$, and $\chi_n : \mathcal{U}_n \rightarrow \mathbb{R}$ the linear form defined recursively, satisfying $\chi_n(\mathbb{1}_{P^\circ}) = (-1)^{\operatorname{dim} P}$ for every polytope $P$.
Deduce Euler's formula $\sum_F (-1)^{\operatorname{dim} F} = 1$ where $F$ ranges over the faces of $P$.

A complex is a non-empty finite set $\mathcal{C}$ of polytopes of $\mathbb{R}^n$ such that for all $P, Q \in \mathcal{C}$, $P \cap Q$ is either empty or simultaneously a face of both $P$ and $Q$. A face of $\mathcal{C}$ is a subset $F \subset |\mathcal{C}|$ that is a face of one of the $P \in \mathcal{C}$.
Show that if $P$ is a polytope of $\mathbb{R}^n$ of dimension $k > 0$, the set of its faces of dimension $k-1$ forms a complex.

A complex is a non-empty finite set $\mathcal{C}$ of polytopes of $\mathbb{R}^n$ such that for all $P, Q \in \mathcal{C}$, $P \cap Q$ is either empty or simultaneously a face of both $P$ and $Q$.
Let $P$ be a polytope of $\mathbb{R}^n$ of dimension $k > 0$ and $x \in P^\circ$. For each face $F$ of dimension $k-1$ of $P$ we denote $F_x = \operatorname{Conv}(F \cup \{x\})$. Show that the family of $F_x$ forms a complex whose realization equals $P$.