Mezard and Parisi recently proposed a method for investigating the thermodynamics of a supercooled liquid around and below its glass transition. This approach assumes the existence of a true phase transition at a (finite) Kauzmann temperature T-K, which is determined, and gives a quantitative estimate of the jump of specific heat DeltaC(v) at T-K. Using the simplest possible scheme, we apply this formalism to a system of pure "soft-spheres" fluid, when the dimension d of the embedding space is lower than 3 (and possibly noninteger). Looking for a lower critical dimension (which we do not find), we assess the mean-field character of the approximation used in the above mentioned approach.