Largest-Area Convex Quadrilateral in a 1.5D Terrain

A $1.5$D terrain is a simple polygon bounded by a line segment $\ell$ and a polygonal chain monotone with respect to the line segment $\ell$. Usually, $\ell$ is chosen aligned to the $x$-axis, and is called the base of the terrain. In this paper, we consider the problem of finding a convex quadrilateral of largest area inside a $1.5$D terrain in $\mathbb{R}^2$. We present an $O(n^2)$ time algorithm for this problem, where $n$ is the number of vertices of the terrain. Finally, we show that the largest area axis-parallel rectangle inside the terrain yields a $\frac{1}{2}$-approximation result to the largest convex quadrilateral problem.