Let $C$ and $D$ be two non-empty convex subsets of $\mathbb{R}^d$ such that $C$ is closed and bounded, $D$ is closed, and $C \cap D = \emptyset$. Show that $D - C$ is a convex closed subset of $\mathbb{R}^d$ not containing $0$.

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 $E$ 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 non-empty subset of $\mathbb{R}^d$. The polar cone of $E$ is defined by $$E^+ := \left\{p \in \mathbb{R}^d : p \cdot x \geq 0, \forall x \in E\right\}$$ and its bi-polar cone by $$E^{++} = (E^+)^+ := \left\{\xi \in \mathbb{R}^d : \xi \cdot p \geq 0, \forall p \in E^+\right\}.$$ Show that $E^+$ and $E^{++}$ are closed convex cones and that $E \subset E^{++}$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \setminus \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}.$$ Let $C$ be the set: $$C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}$$ Show that $C$ is non-empty, convex, closed and bounded.

Show that, for all $\alpha \in \mathbb { R } _ { + } ^ { * } , p _ { \alpha }$ belongs to $E$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges, and $p_\alpha$ is the function $t \mapsto t^\alpha$.

For all $x \in \mathbb { R } _ { + } ^ { * }$ and all $t \in \mathbb { R } _ { + } ^ { * }$, we denote $k _ { x } ( t ) = \mathrm { e } ^ { \min ( x , t ) } - 1$ where $\min ( x , t )$ denotes the smaller of the real numbers $x$ and $t$. Draw a graph of the function $k _ { x }$. Show that $k _ { x }$ belongs to $E$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{M}(E)$ the set of probability measures on $E$. If $\mu \in \mathscr{M}(E)$, we denote by $\mu(x)$ the value $\mu(\{x\})$.
We denote by $\mathscr{P}(E)$ the set of subsets of $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$. If $f \in \mathscr{B}(\mathscr{P}(E), \mathbb{R})$, we set $$\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}.$$
Show that $\mathscr{M}(E)$ is a subset of $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. We denote $\mathscr{P}(E)$ the set of subsets of $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$. If $f \in \mathscr{B}(\mathscr{P}(E), \mathbb{R})$, we set $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Show that $\mathscr{M}(E)$ is a subset of $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.

We denote by $\mathcal{C}([-1,1])$ the vector space of continuous functions from $[-1,1]$ to $\mathbb{C}$ and $\mathcal{T}([-1,1])$ the complex vector subspace of $\mathcal{C}([-1,1])$ generated by the functions $$e_k : t \mapsto e^{i\pi k t}, \quad k \in \mathbb{Z}.$$ Show that $\mathcal{T}([0,1])$ is a subalgebra of $\mathcal{C}([-1,1])$ for the usual multiplication law of functions.

Recall that $\Gamma$ denotes the subgroup of $G_0$ formed by elements $g$ such that $g(V_\mathbb{Z})=V_\mathbb{Z}$.
Show that for all $v,w\in\mathcal{H}$ and all $R\geq 0$, the set $$\{g\in\Gamma \text{ such that } d(gv,w)\leq R\}$$ is finite.

We consider the three vectors $$w_1 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad w_2 = \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ We denote by $T$ the set of vectors $v\in\mathcal{H}$ such that $B(v,w_i)\geq 0$ for all $i\in\{1,2,3\}$.
Show that $T$ is compact and contains $v_0 = \begin{pmatrix}0\\0\\1\end{pmatrix}$.

For all $s>1$, we denote by $P_k(s)$ the subset of $P_k$ formed by vectors $v$ such that $z_v\leq s$.
Show that $P_k(s)$ is finite.

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A_1, \ldots, A_n$ be as defined in Q13.
Justify that for all $p \in \llbracket 1 , n \rrbracket$, we have the inclusion $$A _ { p } \cap \left\{ \left| R _ { n } \right| < x \right\} \subset A _ { p } \cap \left\{ \left| R _ { n } - R _ { p } \right| > 2 x \right\} .$$

Let $f : [a, b] \longrightarrow \mathbf{R}$ be a continuous function. Prove that the restriction $g$ of the function $f$ to the interval $]a, b[$ belongs to the set $\mathscr{D}_{a,b}$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t) \end{array}$$ Using the result of question 10b, deduce that $(\mathbb{C}[A])^*$ is path-connected.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t). \end{array}$$ Using the result of question 10b, deduce that $(\mathbb{C}[A])^*$ is path-connected.

Let $n \geq 1$ be an integer. A non-empty compact subset $P \subset \mathbb{R}^n$ is a polytope if there exist a non-empty finite set $I$ and if for all $i \in I$ there exist a linear form $\ell_i : \mathbb{R}^n \rightarrow \mathbb{R}$ and a real number $a_i \in \mathbb{R}$ such that $P = \{x \in \mathbb{R}^n : \ell_i(x) \leq a_i\ \forall i \in I\}$. A face $F$ of $P$ is a non-empty subset such that there exists $J \subset I$ with $F = F_J = \{x \in P : \ell_j(x) = a_j\ \forall j \in J\}$.
Verify that every face $F$ of $P$ is a polytope and that $\operatorname{dim} F < \operatorname{dim} P$ if $F \neq P$.

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 $\mathbb{R}^n$.

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 $V$.

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 $f_z \in \mathcal{U}_{n-1}$.

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 $P$.

Let $n \geq 1$ be an integer. We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$ and $S_n$ the symmetric group of order $n$. For all $\sigma \in S_n$, we define $P^\sigma \in \mathcal{M}_n(\mathbb{R})$ as follows: for $i, j \in \{1, 2, \ldots, n\}$ we set $P^\sigma_{ij} = 1$ if $j = \sigma(i)$, $P^\sigma_{ij} = 0$ otherwise. Show that $P^\sigma$ is a vertex of $B_n$ for all $\sigma \in S_n$.

Let $n \geq 1$ be an integer. We denote by $\mathbb{C}[[\mathbb{Z}^n]]$ the $\mathbb{C}$-vector space of functions $f : \mathbb{Z}^n \rightarrow \mathbb{C}$. We say that $f \in \mathbb{C}[[\mathbb{Z}^n]]$ is rational if there exists a nonzero $P \in \mathbb{C}[\mathbb{Z}^n]$ such that $Pf \in \mathbb{C}[\mathbb{Z}^n]$. We say that $f$ is torsion if there exists a nonzero $P \in \mathbb{C}[\mathbb{Z}^n]$ such that $Pf = 0$. We denote by $\mathcal{R}$ the $\mathbb{C}$-vector space of rational elements and $\mathcal{T}$ the $\mathbb{C}$-vector space of torsion elements of $\mathbb{C}[[\mathbb{Z}^n]]$.
In the case where $n = 1$, show that the inclusions $0 \subset \mathcal{T} \subset \mathcal{R} \subset \mathbb{C}[[\mathbb{Z}^n]]$ are strict.