The purpose of this paper is twofold. First, we revisit the cent-dian location problem developed by Halpern, considering both the average and maximum distances. We provide a complete characterization of the cent-dians in the case of a tree and present a new algorithm to determine this set in the case of a general network. Second, to deal with distributional justice considerations in the access to the facility, we introduce the concept of generalized center defined as the point which minimizes the difference between maximum and average distances. We show that this point coincides with the center in a tree. An algorithm to find the generalized center in a general network is proposed.