Let $P$ be a $k$-colored set of $n$ points in the plane, $4 \leq k \leq n$. We study the problem of deciding if $P$ contains a subset of four points of different colors such that its Rectilinear Convex Hull has positive area. We show this problem to be equivalent to deciding if there exists a point $c$ in the plane such that each of the open quadrants defined by $c$ contains a point of $P$, each of them having a different color. We provide an $O(n \log n)$-time algorithm for this problem, where the hidden constant does not depend on $k$; then, we prove that this problem has time complexity $\Omega(n \log n)$ in the algebraic computation tree model. No general position assumptions for $P$ are required.

翻译：设$P$为平面上包含$n$个点的$k$色点集，其中$4 \leq k \leq n$。我们研究的问题是：判断$P$是否包含一个由四种不同颜色点构成的子集，使得其正交凸包具有正面积。我们证明该问题等价于判断平面上是否存在一点$c$，使得由$c$定义的四个开象限中各包含$P$的一个点，且这些点颜色互不相同。针对该问题，我们提出了一种时间复杂度为$O(n \log n)$的算法，其中隐藏常数与$k$无关；进而，我们证明该问题在代数计算树模型下具有$\Omega(n \log n)$的时间复杂度下限。该研究不要求点集$P$满足一般位置假设。