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 $n \geq 1$ 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 $F \neq P$.