Locating an Obnoxious Line among Planar Objects

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Given a set P of n points in the plane such that each point has a positive weight, we study the problem of finding an obnoxious line that intersects the convex hull of P and maximizes the minimum weighted Euclidean distance to all points of P. We also consider a variant of this problem whose input is a set of m polygons with totally n vertices in the plane such that each polygon has a positive weight and whose goal is to locate an obnoxious line with respect to the weighted polygons. We improve the previous results for both problems. Our algorithms are based on new geometric observations and interesting algorithmic techniques.