Algorithms for bivariate medians and a fermat-torricelli problem for lines

Given a set S of n points in R 2 , the Oja depth of a point is the sum of the areas of all triangles formed by and two elements of S. A point in R 2 with minimum depth is an Oja median. We show how an Oja median may be computed in O(n log 3 n) time. In addition, we present an algorithm for computing the Fermat-Torricelli points of n lines in O(n) time. These points minimize the sum of weighted distances to the lines. Finally, we propose an algorithm which computes the simplicial median of S in O(n 4) time. This median is a point in R 2 which is contained in the most triangles formed by elements of S.