\[
          \mathring{C} = \bigcap_{\substack{D \text{ droite de } E \\ \text{passant par } x}} \operatorname{int}_D (D \cap C)
        \]

\[
          \dot{C} = \bigcap_D \operatorname{bnd}_D (D \cap C)
        \]

\[
          \overline{C} = \bigcap_D \overline{D \cap C}
        \]

\[
  \begin{array}{ll}
    E_{F'} \subset E_F & \\
    E_{F'} \cap C \subset E_F \cap C & (\text{ou } E_{F'} \cap C \subset E_F) \\
    F' \subset E_F \cap C & (\text{ou } F' \subset E_F) \\
    F' \subset \overline{F} \cap C & (\text{ou } F' \subset \overline{F}) \\
    \overline{F'} \subset \overline{F} &
  \end{array}
\]

\[
  F_x \prec F_y \Leftrightarrow
  \bigl[ x = y \text{ ou } x \neq y \text{ et }
  y \in \operatorname{Int}_{D_{xy}} (D_{xy} \cap C) \bigr]
\]

\[
  F_x = \sup F_{x_i}.
\]

\[
  \Phi = \{ F \in \mathrm{Fac}(C) \mid F \subset C' \}
\]

\[
  A^\circ = \{ x' \in E' \mid \langle x, x' \rangle \geq -1 \ \forall x \in A \}
  = \bigcap_{x \in A} H(x)
  \qquad (\text{où } H(x) = \{ x' \in E' \mid \langle x, x' \rangle \geq -1 \})
\]

\[
  A_1 \subset A_2 \Longleftrightarrow A_2^\circ \subset A_1^\circ
\]

\[
  \overline{\mathrm{Env}}(A_1, A_2) = A_1^\circ \cap A_2^\circ
\]

\[
  (A_1 \cap A_2)^\circ = \overline{\mathrm{Env}}(A_1, A_2)
\]

\[
  U = \{ \lambda_1 e_1 + \lambda_2 e_2 \mid \lambda_1, \lambda_2 > 0 \}
\]

\[
  C = C'^\circ, \quad C' = C^\circ.
\]

\[
  C = C_1 \cap H
  \qquad (H \text{ demi-espace fermé, limité par l'hyperplan } H_0)
\]

\[
  \dim F + \dim F^{0} = n - 1 \qquad (n = \dim E = \dim E')
\]

\[
  E_{\varphi(F) = F'} = (E_F)^{0}
\]

\[
  \dim F_2 = \dim F_1 + 1
\]

\[
  \Phi \subset \mathrm{Espaff}(E)
\]

\[
      \Phi \setminus \{\emptyset\} \longrightarrow \Gamma_{\mathrm{part}}(B/I)
    \]

\[
  T \simeq \prod T_i \qquad (T_i = T/\mathbb{H}_i)
\]

\[
  T_i = \operatorname{Ker}(k^{B_i} \xrightarrow{\varepsilon} k), \quad
  E_i = \varepsilon^{-1}(1)
\]

\[
  T = \operatorname{Ker}(k^B \xrightarrow{\mathrm{tr}} k^I), \qquad
  E = \mathrm{tr}^{-1}(1)
\]

\[
  C_n \longrightarrow \mathcal{D}_n \qquad \text{par} \quad
  \bigl(E, (E_i), (B_i)_{i\in I}\bigr) \longmapsto \Bigl\{\textstyle\bigcup B_i, \ldots\Bigr\}
\]

\[
  \mathcal{D}_n \longrightarrow C_n \quad \text{en prenant} \quad
  E = \coprod_{i\in I} B_i \wedge_{\{\pm 1\}} k
\]

\[
  \omega_E \simeq \omega_I \wedge \bigwedge_{i\in I} B_i .
\]

\[
  \omega_E \simeq \omega_I \wedge \bigwedge_{i\in I} \omega_{E_i}
\]

\[
  \omega_{E_i} \simeq \struck{\ill{}}\, B_i .
\]

\[
  F_0 \lneq F_1 \lneq \cdots \lneq F_n = F
\]

\[
  \Phi_0 \quad \Phi_1 \quad \ldots \quad \Phi_{n-1}\struck{\ill{}}
  \qquad (n = \dim C)
\]

\[
  \sum_{b\in B} \lambda_b . b \quad \text{avec} \quad
  \lambda_{-b} = -\lambda_b, \quad \lambda_{\sigma(j)} = +1, \quad \text{et}
\]

\[
  \underset{-1}{\bullet} \!\!\!\longrightarrow\!\!\!
  \underset{\lambda_b}{\bullet} \!\!\!\longrightarrow\!\!\!
  \underset{+1 = \mu_b}{\bullet}
  \qquad \Longrightarrow \qquad \lambda_b = \mu_b \quad \text{donc}
\]

\[
  C = \prod_{i\in I} B_i \qquad \bigl(B_i = p^{-1}(i)\bigr)
\]

\[
  C_b \cap C_{b'} = \emptyset \iff p(b) = p(b') .
\]

\[
  \bigl(p(b) = p(b')\bigr) \iff \bigl(b = b' \ \text{ou}\ C_b \cap C_{b'} = \emptyset\bigr)
\]

\[
      C_n \subset \mathbb{R}^n, \qquad
      C_n = \bigl\{\, x = (x_i)_{1\leq i\leq n} \in \mathbb{R}^n \bigm| |x_i| \leq 1 \,\bigr\}
    \]

\[
      W_n = \operatorname{Aut}_{\text{affin.}}(\mathbb{R}^n, C_n)
    \]

\[
  \operatorname{Aut}_{\text{aff}}(\mathbb{R}^n, C_n) = \operatorname{Aut}_{\text{vect}}(\mathbb{R}^n, B_n)
  \xrightarrow{\ \sim\ } \operatorname{Aut}(B_n, \uncertain{\sigma_n})
\]

\[
  \underset{x'}{\bullet}\!-\!\underset{x}{\bullet}\longrightarrow y \longrightarrow y'
\]

\[
  R(x,y) : \ (x = y) \ \text{\emph{ou}} \ \bigl(x \text{ ni } y \text{ ne sont extrémités de } C \cap D_{xy}\bigr)
\]

\[
  \underset{x'}{\bullet}\,\underset{x}{\bullet} \ \ldots \ \underset{y''}{\bullet}\,\underset{y}{\bullet}\,\underset{y'}{\bullet} \ \ldots \ \underset{z}{\bullet}\,\underset{z'}{\bullet}
\]

\[
  A^{\wedge} = \operatorname{Im}(A^{\llcorner\!\to} \to S \times F)
  = \text{rel. d'incidence entre $S$ et $F$}
\]

\begin{align*}
  A^{\llcorner\!\to} &\simeq \widetilde{S} \times_S \vec{A} \simeq \struck{\ill{}}\, A^{\uparrow} \times_F \widetilde{F} \\
  &\simeq \vec{A} \times_A A^{\uparrow} \\
  &\simeq \vec{A} \times_A \widetilde{A} \simeq \widetilde{A} \times_A A^{\uparrow} \\
  &\simeq \widetilde{S} \times_S A^{\wedge} \simeq A^{\wedge} \times_F \widetilde{F}
\end{align*}