To gain insight into the basic variables to be utilized in constructing improved energy density functionals, various models of electron confinement will be discussed, in which analytic forms of either kinetic or exchange energy functionals can be derived. Among these models are (i) the Bardeen infinite barrier model of a surface (ii) the semi-infinite jellium model and (iii) an electron, confined by an infinite barrier but in a homogeneous electric field. In (i), an explicit non-local relation between kinetic and exchange energies can also be exhibited. Finally a relativistic generalization of what is essentially the Bardeen model relation between kinetic energy and density profile has been effected: this offers considerable simplification due to the fact that Dirac four-component wave functions are thereby by-passed. ©2001 American Institute of Physics.