The aim of this article is to continue to develop the theory of generalized weighted means of functions relative to weight functions in Euclidean n-space (Glaser, 1990) by proving a generalization of the one-dimensional Cashwell-Everett linear ordering theorem (Cashwell-Everett (1969). These generalized means lie in an n-dimensional cone which shrinks down to a line for the special case n = 1. This theorem has many interesting consequences which could be the subject of future articles.