The Legendre transform of an (approximate) expression for the ground-state energy E-0(eta,g) of an N-electron system yields the one-matrix functional V-ee[Gamma(x('),x)] for the electron-electron repulsion energy that is given by the function V-ee(n;g) of the occupation numbers n pertaining to Gamma(x('),x) and the two-electron repulsion integrals g computed in the basis of the corresponding natural spinorbitals. Extremization of the electronic energy functional, which is a sum of V-ee[Gamma(x('),x)] and the contraction of Gamma(x('),x) with the core Hamiltonian, produces the (approximate) ground-state energy even if E-0(eta,g) itself is not variational. Thanks to this property, any electron correlation formalism can be reformulated in the language of the density matrix functional theory. Ten conditions that have to be satisfied by V-ee(n;g) uncover several characteristics of V-ee[Gamma(x('),x)]. In particular, when applied in conjunction with the homogeneity property, the condition of volume extensivity imposes stringent constraints upon the possible dependence of V-ee(n;g) on g.