Let $\mathcal{U}_n$ be the vector subspace of functions $\mathbb{R}^n \rightarrow \mathbb{R}$ generated by indicator functions of polytopes of $\mathbb{R}^n$, and $\chi_n : \mathcal{U}_n \rightarrow \mathbb{R}$ the linear form defined recursively. For a polytope $P$ of $\mathbb{R}^n$, let $P^\circ$ denote its relative interior.
Show that for every polytope $P$ of $\mathbb{R}^n$, $\mathbb{1}_{P^\circ} \in \mathcal{U}_n$ and $\chi_n(\mathbb{1}_{P^\circ}) = (-1)^{\operatorname{dim} P}$.