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The greatest convex function lying below a given function, when such an envelope is considered in the specified domain. A convex combination of phase compositions and their Gibbs free energies constructs candidate mixtures; the equilibrium free energy is their lower convex envelope. For two differentiable coexisting branches, the supporting line is a common tangent.

For a mixture with phase compositions $x_1,x_2$, its molar Gibbs free energy is the straight-line combination $g=(1-f)g_1(x_1)+f{g}_2(x_2)$ at overall $x=(1-f)x_1+f{x}_2$. Minimizing over phase compositions gives equal slopes and equal tangent intercepts: the common-tangent construction expresses equality of both component chemical potentials. The equilibrium curve is the lower convex envelope of the candidate phase curves. Phase fractions follow the lever rule.