Linear approximation of simple objects

Let S be a set of m convex polygons in the plane with a total number of n vertices. Each polygon has a positive weight. This paper presents algorithms to solve the weighted minmax approximation and the weighted minsum approximation problems. For the first problem, a line minimizing the maximum weighted distance to the polygons can be found in O(n² log n) time and O(n²) space. The time and space complexities can be reduced to O(n log n) and O(n), respectively, when the weights are equal. For the second problem, a line minimizing the sum of the weighted distances to the polygons can be found in O(n² log n) time and O(n) space. For both problems, we also obtain similar results for sets of n circles or line segments.