The central role of the shape function sigma((r) under bar) from the density functional theory (DFT), the ratio of the electron density rho((r) under bar) and the number of electrons N of the system (density per particle), is investigated. Moreover, its relationship with DFT based reactivity indices is established. In the first part, it is shown that an estimate for the chemical hardness can be obtained from the long range behavior of the shape function and its derivative with respect to the number of electrons at a fixed external potential. Next, the energy of the system is minimized with the constraint that the shape function should integrate to unity; the associated Lagrange multiplier is shown to be related to the electronic chemical potential mu of the system. Finally, the importance of the shape function for both molecular structure, reactivity, and similarity is outlined. (C) 2004 American Institute of Physics.