Let $\Pi$ be a complete polyhedral complex in $N_{\mathbb{R}}$. Then $\operatorname{rec}(\Pi)$ and $\operatorname{c}(\Pi)$ are complete conic polyhedral complexes in $N_{\mathbb{R}}$ and $N_{\mathbb{R}}\times\mathbb{R}_{\geq 0}$, respectively. If, in addition, $\Pi$ is strongly convex, then both $\operatorname{rec}(\Pi)$ and $\operatorname{c}(\Pi)$ are fans.