Computing a maxian point of a simple rectilinear polygon

Let P be a simple rectilinear polygon with n vertices, endowed with rectilinear metric. Let us assign to points Xl,..., xk E P positive weights ~1, . . . , wk. The median problem consists in the computing the point minimizing the total weighted distances to the given points. We present an 0 (n + k log n ) algorithm for solving this median problem. If all weighted points are vertices of a polygon P, then the running time becomes O(n + k).