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.

翻译：$1.5$维地形是一个由线段$\ell$和一条相对于线段$\ell$单调的多边形链所围成的简单多边形。通常，$\ell$被选择与$x$轴对齐，并称为地形的基底。本文研究在$\mathbb{R}^2$中的$1.5$维地形内寻找最大面积凸四边形的问题。我们提出了一个时间复杂度为$O(n^2)$的算法来解决该问题，其中$n$是地形顶点的数量。最后，我们证明了地形内最大面积的轴对齐矩形可为最大凸四边形问题提供$\frac{1}{2}$近似解。