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 $K \in \mathbb{N}^{\star}$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$; (H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent.
In the special case $[a,b] = [0,1]$, justify that the series $\sum f_n^{(k)}$ converges normally on $[a,b]$ for all $k \in \llbracket 0, K-1 \rrbracket$.

Let $n \in \mathbb{N}^*$ and $W$ be a monic polynomial of degree $n$. The objective of this subsection is to show that $$\sup _ { x \in [ - 1,1 ] } | W ( x ) | \geqslant \frac { 1 } { 2 ^ { n - 1 } }.$$ Show that $\sup _ { x \in [ - 1,1 ] } \left| T _ { n } ( x ) \right| = 1$. Deduce a monic polynomial of degree $n$ achieving equality in the above inequality.

We denote $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. Show that $\rho(A) \leqslant \max_{U \in C} \left| U^\top A U \right|$.

Let $k \in \mathbb{N}^*, (p_1, \ldots, p_k) \in (\mathbb{R}^d)^k$ and $(b_1, \ldots, b_k) \in \mathbb{R}^k$ such that $$A := \left\{x \in \mathbb{R}^d : p_i \cdot x \leqslant b_i, i = 1, \ldots, k\right\}$$ is non-empty. Show that $A$ is convex and closed. Let $x \in A$, let $I(x) := \{i \in \{1, \ldots, k\} : p_i \cdot x = b_i\}$, show that $$x \in \operatorname{Ext}(A) \Longleftrightarrow \operatorname{rank}\left(\{p_i, i \in I(x)\}\right) = d$$ deduce that $\operatorname{Ext}(A)$ is a finite set (possibly empty) whose cardinality is at most $2^k$.

Let $k \in \mathbb{N}^*, (p_1, \ldots, p_k) \in (\mathbb{R}^d)^k$ and $(b_1, \ldots, b_k) \in \mathbb{R}^k$ such that $$A := \left\{x \in \mathbb{R}^d : p_i \cdot x \leqslant b_i, i = 1, \ldots, k\right\}$$ is non-empty. Show that $A$ is convex and closed. Let $x \in A$, let $I(x) := \left\{i \in \{1, \ldots, k\} : p_i \cdot x = b_i\right\}$, show that $$x \in \operatorname{Ext}(A) \Longleftrightarrow \operatorname{rank}\left(\{p_i, i \in I(x)\}\right) = d$$ deduce that $\operatorname{Ext}(A)$ is a finite set (possibly empty) whose cardinality is at most $2^k$.

Let $K \in \mathbb{N}^{\star}$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$; (H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent.
Treat the question of showing that $\sum f_n^{(k)}$ converges normally on $[a,b]$ for all $k \in \llbracket 0, K-1 \rrbracket$ in the general case of a segment $[a,b]$ with $a < b$. One may examine $f_n \circ \sigma$ where $\sigma : [0,1] \rightarrow [a,b]$ is defined by $\sigma(t) = (1-t)a + tb$ for all $t \in [0,1]$.

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 $\rho(A) = \max_{U \in C} \left| U^\top A U \right|$.

Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Let $p \in \mathbb{R}^d$, set $$K_p := \{x \in K : p \cdot x \leqslant p \cdot y, \forall y \in K\}.$$ Show that $K_p$ is non-empty, convex and closed and that $\operatorname{Ext}(K_p) \subset \operatorname{Ext}(K)$.

Let $K \in \mathbb{N}^{\star}$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ satisfying hypotheses (H1) and (H2). According to the result of the previous question, we set $F_k(x) = \sum_{n=0}^{+\infty} f_n^{(k)}(x)$ for all $x \in [a,b]$. Prove that $F_0$ is of class $\mathcal{C}^K$ on $[a,b]$ and that $F_0^{(k)} = F_k$ for all $k \in \llbracket 1, K \rrbracket$.

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 $k \in \llbracket 0 , n \rrbracket$, $z _ { k } = \cos \left( \frac { k \pi } { n } \right)$. In this question, we prove $$\sup _ { x \in [ - 1,1 ] } | W ( x ) | \geqslant \frac { 1 } { 2 ^ { n - 1 } }$$ by contradiction.

• If we assume that $\sup _ { x \in [ - 1,1 ] } | W ( x ) | < \frac { 1 } { 2 ^ { n - 1 } }$, show that, for all $k \in \llbracket 0 , n - 1 \rrbracket , Q \left( z _ { k } \right) Q \left( z _ { k + 1 } \right) < 0$.
• Deduce a contradiction and conclude.

Let $A \in \mathcal{S}_n(\mathbb{R})$ and $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. We further assume that the eigenvalues of $A$ are all positive. Show then that $\rho(A) = \max_{U \in C} \left( U^\top A U \right)$.

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 $n \in \mathbb{N}^*$, $W$ 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 $k \in \llbracket 0 , n \rrbracket$, $$\frac { Q \left( z _ { k } \right) } { \prod _ { \substack { j = 0 \\ j \neq k } } ^ { n } \left( z _ { k } - z _ { j } \right) } \geqslant 0.$$

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 $\mathcal{S}_n(\mathbb{R})$.

Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Show that $K = \operatorname{co}(\operatorname{Ext}(K))$.

Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Show that $K = \operatorname{co}(\operatorname{Ext}(K))$.

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 assume that $\sup _ { x \in [ - 1,1 ] } | W ( x ) | = \frac { 1 } { 2 ^ { n - 1 } }$. Deduce that $Q = 0$, then that $W = \frac { 1 } { 2 ^ { n - 1 } } T _ { n }$.
One may consider the sum of the inequalities from the previous question and exploit question 6 applied to suitable data.

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

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 $r$ be a real-valued function of class $\mathcal { C } ^ { n }$ on $I = [a,b]$ and vanishing at $n + 1$ distinct points of $I$. Show that there exists $c \in I$ such that $r ^ { ( n ) } ( c ) = 0$.

Let $E$ be a non-empty subset of $\mathbb{R}^d$. Using the definitions of $E^+$ and $E^{++}$ from question 16, show that $E = E^{++}$ if and only if $E$ is a closed convex cone.

Let $E$ be a non-empty subset of $\mathbb{R}^d$. With $E^+$ and $E^{++}$ as defined in question 16, show that $E = E^{++}$ if and only if $E$ is a closed convex cone.

Let $n$ be a nonzero natural integer, $I = [a,b]$ with $a < b$, and $a_1 < \cdots < a_n$ distinct real numbers in $I$. Let $W = \prod_{i=1}^n (X - a_i)$ and let $f$ be a real-valued function of class $\mathcal { C } ^ { n }$ on $I$. Let $P = \Pi ( f )$ be the interpolation polynomial of $f$ associated with the real numbers $a _ { 1 } , \ldots , a _ { n }$, defined by $$\Pi ( f ) = \sum _ { i = 1 } ^ { n } f \left( a _ { i } \right) L _ { i }.$$ For all $x \in I$, show that there exists $c \in I$ such that $$f ( x ) - P ( x ) = \frac { f ^ { ( n ) } ( c ) } { n ! } W ( x ).$$ For $x$ distinct from the $a _ { i }$, one may consider the function $r$ defined on $I$ by $$r ( t ) = f ( t ) - P ( t ) - K W ( t )$$ where the real number $K$ is chosen so that $r ( x ) = 0$.