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 $$p \cdot x \leqslant p \cdot y, \forall y \in A.$$

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.

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.

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 a countably infinite set where the $x_i$ are pairwise distinct elements. 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 $\|\cdot\|$ defines a norm on the vector space $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.

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 $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. 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 $\|\cdot\|$ defines a norm on the vector space $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.

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 $E$ 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 $\operatorname{Ext}(\operatorname{co}(E)) \subset E$.

Let $A = \operatorname{co}(E)$ where $E$ is the subset of $\mathbb{R}^3$ defined by $$E = \{(0,0,1),(0,0,-1)\} \cup \{(1+\cos(\theta), \sin(\theta), 0), \theta \in [0, 2\pi]\}$$ show that $\operatorname{Ext}(A)$ is non-empty and is not closed.

Let $A = \operatorname{co}(E)$ where $E$ is the subset of $\mathbb{R}^3$ defined by $$E = \{(0,0,1),(0,0,-1)\} \cup \{(1 + \cos(\theta), \sin(\theta), 0), \theta \in [0, 2\pi]\}$$ show that $\operatorname{Ext}(A)$ is non-empty and is not closed.

Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Show that $\operatorname{Ext}(K)$ is non-empty (one may reduce to the case where $0 \in K$ and reason on the dimension of $K$).

Let $u \in \operatorname { Symp } _ { \omega } ( E )$ be a symplectic endomorphism of $E$. Let $\lambda , \mu$ be real eigenvalues of $u$, and let $E _ { \lambda } ( u ) , E _ { \mu } ( u )$ be the associated eigenspaces. Show that, if $\lambda \mu \neq 1$, then the subspaces $E _ { \lambda } ( u )$ and $E _ { \mu } ( u )$ are $\omega$-orthogonal, that is:
$$\forall x \in E _ { \lambda } ( u ) , \quad \forall y \in E _ { \mu } ( u ) , \quad \omega ( x , y ) = 0$$

Let $u \in \operatorname { Symp } _ { \omega } ( E )$ be a symplectic endomorphism of $E$. Let $\lambda , \mu$ be real eigenvalues of $u$, and let $E _ { \lambda } ( u ) , E _ { \mu } ( u )$ be the associated eigenspaces. Show that, if $\lambda \mu \neq 1$, then the subspaces $E _ { \lambda } ( u )$ and $E _ { \mu } ( u )$ are $\omega$-orthogonal, that is:
$$\forall x \in E _ { \lambda } ( u ) , \quad \forall y \in E _ { \mu } ( u ) , \quad \omega ( x , y ) = 0$$

Let $C _ { 1 } , \ldots , C _ { n }$ be $n$ column matrices in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$, with $C _ { 1 }$ non-zero.
Prove that, if the family $\left( C _ { 1 } , \ldots , C _ { n } \right)$ is linearly dependent, then there exists a unique $j \in \llbracket 1 , n - 1 \rrbracket$ such that $$\left\{ \begin{array} { l } \left( C _ { 1 } , \ldots , C _ { j } \right) \text { is linearly independent } \\ C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right) \end{array} \right.$$

Let $n \in \mathbb{N}^*$, $W$ be a monic polynomial of degree $n$, $Q = \frac { 1 } { 2 ^ { n - 1 } } T _ { n } - W$, and for all $k \in \llbracket 0 , n \rrbracket$, $z _ { k } = \cos \left( \frac { k \pi } { n } \right)$. We now assume that $\sup _ { x \in [ - 1,1 ] } | W ( x ) | = \frac { 1 } { 2 ^ { n - 1 } }$. Show that, for all $k \in \llbracket 0 , n \rrbracket$, $$\frac { Q \left( z _ { k } \right) } { \prod _ { \substack { j = 0 \\ j \neq k } } ^ { n } \left( z _ { k } - z _ { j } \right) } \geqslant 0.$$

The functions $p _ { \alpha }$ are defined by $p_\alpha : t \mapsto t^\alpha$ for $\alpha \in \mathbb{R}_+^*$, and the inner product on $E$ is $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Is the family $\left( p _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ an orthogonal family of $E$?

We denote by $\mathcal{R}_{n}$ the set of rational functions with no pole in $\mathbb{U}$ of the form $\frac{P}{Q}$ where $P$ and $Q$ are two elements of $\mathbf{C}_{n}[X]$.
Let $F \in \mathcal{R}_{n}$, $P$ and $Q$ be two elements of $\mathbf{C}_{n}[X]$ satisfying $F = \frac{P}{Q}$ and $$\forall z \in \mathbb{U}, \quad Q(z) \neq 0$$ For $t \in [-\pi, \pi]$, we set $$f(t) = F\left(e^{it}\right) = g(t) + ih(t) \quad \text{where} \quad (g(t), h(t)) \in \mathbf{R}^{2}$$ For $u \in [-\pi, \pi]$, we define a function $f_{u}$ from $[-\pi, \pi]$ to $\mathbf{R}$ by $$\forall t \in [-\pi, \pi], \quad f_{u}(t) = g(t)\cos(u) + h(t)\sin(u) = \operatorname{Re}\left(e^{-iu}F\left(e^{it}\right)\right) = \operatorname{Re}\left(e^{-iu}f(t)\right).$$
In this question, we fix $u \in [-\pi, \pi]$ and assume that $f_{u}$ is not constant. We also fix $y \in \mathbf{R}$. Using if necessary the expression of $f_{u}(t)$ as the real part of $e^{-iu}F\left(e^{it}\right)$ and Euler's formula for the real part, determine $S \in \mathbf{C}_{2n}[X]$ such that $$\forall t \in [-\pi, \pi], \quad f_{u}(t) = y \Longleftrightarrow S\left(e^{it}\right) = 0.$$ Deduce that the set $f_{u}^{-1}(\{y\}) \cap [-\pi, \pi[$ is finite with cardinality bounded by $2n$.

Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. We denote by $\mathbb{E}(X)$ the expectation of a real random variable $X$. Let $\mathscr{P}(E)$ be the set of subsets of $E$ and $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
Show that for all random variables $X$ and $Y$ on $(\Omega, \mathscr{A}, P)$ and for every subset $A$ of $E$: $$|\mu_X(A) - \mu_Y(A)| \leqslant \mathbb{E}\left(|\mathbb{1}_{\{X \in A\}} - \mathbb{1}_{\{Y \in A\}}|\right)$$ and deduce that $\|\mu_X - \mu_Y\| \leqslant P(X \neq Y)$, where $\{X \neq Y\} = \{\omega \in \Omega \text{ such that } X(\omega) \neq Y(\omega)\}$.

Let $E$ be an infinite countable subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Show that for all random variables $X$ and $Y$ on $(\Omega, \mathscr{A}, \mathbf{P})$ and for all subset $A$ of $E$: $$\left|\mu_X(A) - \mu_Y(A)\right| \leqslant \mathbf{E}\left(\left|\mathbf{1}_{\{X \in A\}} - \mathbf{1}_{\{Y \in A\}}\right|\right)$$ and deduce that $\left\|\mu_X - \mu_Y\right\| \leqslant \mathbf{P}(X \neq Y)$, where $\{X \neq Y\} = \{\omega \in \Omega \text{ such that } X(\omega) \neq Y(\omega)\}$.

Let $A \in \mathcal{S}_n(\mathbb{R})$ and $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. Prove that the application $\rho$ defines a norm on $\mathcal{S}_n(\mathbb{R})$.

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Show that for all $k \in \mathbf { N }$, the random variable $X ^ { k }$ has finite expectation. Show that $\Phi _ { X }$ is of class $\mathcal { C } ^ { \infty }$ on $\mathbf { R }$ and that $\Phi _ { X } ^ { ( k ) } ( 0 ) = i ^ { k } \mathbf { E } \left( X ^ { k } \right)$ for all $k \in \mathbf { N }$.