Covering Polygons by Min-Area Convex Polygons
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Given a set of disjoint simple polygons σ_1, ..., σ_n, of total complexity N, consider a convexification process that repeatedly replaces a polygon by its convex hull, and any two (by now convex) polygons that intersect by their common convex hull. This process continues until no pair of polygons intersect. We show that this process has a unique output, which is a cover of the input polygons by a set of disjoint convex polygons, of total minimum area. Furthermore, we present a near linear time algorithm for computing this partition. The more general problem of covering a set of N segments (not necessarily disjoint) by min-area disjoint polygons can also be solved in near linear time.
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