# Maximum overlap area of a convex polyhedron and a convex polygon under translation

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 ∈ℝ^3 maximizing the overlap area |P ∩ (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.

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