  The inverse Henderson problem refers to the determination of the pair
potential which specifies the interactions in an ensemble of classical
particles in continuous space, given the density and the equilibrium pair
correlation function of these particles as data. For a canonical ensemble in a
bounded domain it has been observed that this pair potential minimizes a
corresponding convex relative entropy functional, and that the Newton iteration
for minimizing this functional coincides with the so-called inverse Monte Carlo
(IMC) iterative scheme. In this paper we show that in the thermodynamic limit
analogous connections exist between the specific relative entropy introduced by
Georgii and Zessin and a proper formulation of the IMC iteration in the full
space. This provides a rigorous variational framework for the inverse Henderson
problem, valid within a large class of pair potentials, including, for example,
Lennard-Jones type potentials.
  It is further shown that the pressure is strictly convex as a function of the
pair potential and the chemical potential, and that the specific relative
entropy at fixed density is a strictly convex function of the pair potential.
At a given reference potential and a corresponding density in the gas phase we
determine the gradient and the Hessian of the specific relative entropy, and we
prove that the Hessian extends to a symmetric positive semidefinite quadratic
functional in the space of square integrable perturbations of this potential.
