  Interrelation of the Coleman's representabilty theory for 1-density operators
and abstract algebraic form of the Hohenberg-Kohn theorem is studied in detail.
Convenient realization of the Hohenberg-Kohn set of classes of 1-electron
operators and the Coleman's set of ensemble representable 1-density operators
is presented. Dependence of the Hohenberg-Kohn class structure on the boundary
properties of the ground state 1-density operator is established and is
illustrated on concrete simple examples. Algorithm of restoration of many
electron determinant ensembles from a given 1-density diagonal is described.
Complete description of the combinatorial structure of Coleman's polyhedrons is
obtained.
