Let $\mathcal{U}_n$ be the vector subspace of functions $\mathbb{R}^n \rightarrow \mathbb{R}$ generated by indicator functions of polytopes of $\mathbb{R}^n$. For a polytope $P$ of $\mathbb{R}^n$, the relative interior $P^\circ$ is defined as $P^\circ = \{x \in P : \ell_i(x) = a_i \Leftrightarrow i \in S_F\}$ where $S_F = \{i \in I, \ell_i(x) = a_i\ \forall x \in F\}$. Show that for every polytope $P$ of $\mathbb{R}^n$ and for all $x \in P \backslash P^\circ$, there exists a face $F \subset P$ such that $F \neq P$ and $x \in F$.
Let $\mathcal{U}_n$ be the vector subspace of functions $\mathbb{R}^n \rightarrow \mathbb{R}$ generated by indicator functions of polytopes of $\mathbb{R}^n$. For a polytope $P$ of $\mathbb{R}^n$, the relative interior $P^\circ$ is defined as $P^\circ = \{x \in P : \ell_i(x) = a_i \Leftrightarrow i \in S_F\}$ where $S_F = \{i \in I, \ell_i(x) = a_i\ \forall x \in F\}$.
Show that for every polytope $P$ of $\mathbb{R}^n$ and for all $x \in P \backslash P^\circ$, there exists a face $F \subset P$ such that $F \neq P$ and $x \in F$.