The Inverse Weber Problem on the Plane and the Sphere

Weber's inverse problem in the plane is to modify the positive weights associated with n fixed points in the plane at minimum cost, ensuring that a given point a priori becomes the Euclidean weighted geometric median. In this paper, we investigate Weber's inverse problem in the plane and generalize it to the surface of the sphere. Our study uses a subspace orthogonal to a subspace generated by two vectors X and Y associated with the given points and weights. The main achievement of our work lies in determining a vector perpendicular to the vectors X and Y, in Rn; which is used to determinate a solution of Weber's inverse problem. In addition, lower bounds are obtained for the minimum of the Weber function, and an upper bound for the difference of the minimal of Weber's direct and inverse problems. Examples of application at the plane and unit sphere are given.