Let $P$ be a convex polyhedron and $Q$ be a convex polygon with $n$ vertices in total in three-dimensional space. We present a deterministic algorithm that finds a translation vector $v \in \mathbb{R}^3$ maximizing the overlap area $|P \cap (Q + v)|$ in $O(n \log^2 n)$ time. We then apply our algorithm to solve two related problems. We give an $O(n \log^3 n)$ time algorithm that finds the maximum overlap area of three convex polygons with $n$ vertices in total. We also give an $O(n \log^2 n)$ time algorithm that minimizes the symmetric difference of two convex polygons under scaling and translation.

翻译：设$P$为凸多面体，$Q$为三维空间中顶点总数为$n$的凸多边形。我们提出一种确定性算法，可在$O(n \log^2 n)$时间内找到平移向量$v \in \mathbb{R}^3$，使得重叠面积$|P \cap (Q + v)|$最大化。随后，我们将该算法应用于解决两个相关问题：给出一个$O(n \log^3 n)$时间的算法，用于寻找三个顶点总数为$n$的凸多边形的最大重叠面积；并给出一个$O(n \log^2 n)$时间的算法，用于在缩放和平移下最小化两个凸多边形的对称差。