The binding of a single positron to a many-electron system (in the field of a fixed nuclear configuration) is viewed in terms of positron and electron densities. Following the treatments due to Levy and co-workers [Phys. Rev. A 32, 2010 (1985); 30, 2745 (1984)], certain general integral conditions and also an exact differential equation for the positron density are derived. It is demonstrated further that within a density-functional approach, decent estimates to the positron affinity for negative ions can be obtained by invoking the Slater transition state concept. Finally, the redistribution in the electron part brought about by transiently binding the positron to the negative ion is discussed at the density-matrix level.