New relations among reactivity descriptors are provided within the recently introduced modified isomorphic ensemble of density functional theory. In addition, expressions for the softness and hardness kernel are derived in the canonical, grand canonical, isomorphic and grand isomorphic ensemble. There results a new definition for the local hardness, eta(r)=[partial derivative u(r)/partial derivative N](sigma) = eta(sigma)-g(r), where g(r) = [partial derivative v(r)/partial derivative N](sigma) and sigma = rho/N is the shape factor. This identifies the local hardness as a function measuring the response of the system's external potential to a perturbation in electron number at a constant shape factor. Furthermore, it is shown that one cannot represent both local softness and local hardness unambiguously with one representation. (C) 1997 American Institute of Physics.