A complex $\mathcal{C}$ is a non-empty finite set of polytopes of $\mathbb{R}^n$ such that for all $P, Q \in \mathcal{C}$, $P \cap Q$ is either empty or simultaneously a face of both $P$ and $Q$. We denote $\chi(\mathcal{C}) = \sum_F (-1)^{\operatorname{dim} F}$ where $F$ ranges over the faces of $\mathcal{C}$.
Show that every complex $\mathcal{C}$ whose realization is convex satisfies $\chi(\mathcal{C}) = 1$.