Convex Hull for Planar H-Polyhedra

Suppose $\langle A_i, \vec{c}_i \rangle$ are planar (convex) H-polyhedra, that is, (unknown variable A_i) \in \mathbb{R}^{n_i \times 2}$ and $\vec{c}_i \in \mathbb{R}^{n_i}$. Let (unknown variable P_i) = \{ \vec{x} \in \mathbb{R}^2 \mid A_i\vec{x} \leq \vec{c}_i \}$ and (unknown variable n) = n_1 + n_2$. We present an (unknown variable O)(n \log n)$ algorithm for calculating an H-polyhedron $\langle A, \vec{c} \rangle$ with the smallest (unknown variable P) = \{ \vec{x} \in \mathbb{R}^2 \mid A\vec{x} \leq \vec{c} \}$ such that (unknown variable P)_1 \cup P_2 \subseteq P$.