A convexity principle for interacting gases

A new set of inequalities is introduced, based on a novel but natural interpolation between Borel probability measures on R-d. Using these estimates in lieu of convexity or rearrangement inequalities, the existence and uniqueness problems are solved for a family of attracting gas models. In these models, the gas interacts with itself through a force which increases with distance and is governed by an equation of state P = P(rho) relating pressure to density. P(rho)/rho((d-1)/d) is assumed non-decreasing for a ti-dimensional gas. By showing that the internal and potential energies for the system are convex Functions of the interpolation parameter, an energy minimizing state - unique up to translation - is proven to exist. The concavity established for \\rho(t)\\(-p/d) as a function of t is an element of [0, 1] generalizes the Brunn-Minkowski inequality from sets to measures. (C) 1997 Academic Press.