Within the density functional formalism, we introduce the shape chemical potential mu(i)(n) for subsystems, which in the limiting case of point subsystems, is a local chemical potential mu(n)(r). It describes the electron withdrawing/donating ability of specified density fragments. The shape chemical potential does not equalize between subsystems, and provides a powerful new method to identify and describe local features of molecular systems. We explore the formal properties of mu(i)(n) especially with respect to discontinuities, and reconcile our results with Sanderson＇s principle. We also perform preliminary calculations on model systems of atoms in molecules,and atomic shell structure, demonstrating how mu(i)(n) and mu(n)(r), identify and characterize chemical features as regions of different shape chemical potential. We present arguments that shell structure, and other chemical features, are not ever obtainable within Thomas-Fermi-type theories.