  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.
