A complex is a non-empty finite set $\mathcal{C}$ 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 $Q$.
Let $P$ be a polytope of $\mathbb{R}^n$ of dimension $k > 0$ and $x \in P^\circ$. For each face $F$ of dimension $k-1$ of $P$ we denote $F_x = \operatorname{Conv}(F \cup \{x\})$. Show that the family of $F_x$ forms a complex whose realization equals $P$.