We present a method of constructing the phase diagram at low temperatures, using the low temperature expansions. We consider spin Iattice systems described by a Hamiltonian with a d-dimensional perturbation space. We prove that there is a one-one correspondence between subsets of the phase diagram and extremal elements of some family of convex sets. We also solve a linear programming problem of the phase diagram for a set of affine functionals. / Ph. D.