A novel approach to energy-density-functional theory, based on point transformations in coordinate three-dimensional space, is advanced. These point transformations are discussed in the context of one-electron densities and many-electron wave functions. In particular, the group properties of these transformations are discussed and it is shown by incorporating some of the topological properties of the one-electron density, as discussed by Bader et al., that the n-representability conditions for densities may be reformulated. This is done in this paper for atoms. As a simple application of this method, the local-density approximation is obtained by computing the point transformation for a slowly varying density. Also the relationship between this method and the one based in equidensity orbitals is discussed.