The concavity of Gibbs potiential Excuse me, I have a question about the concavity of Gibbs potential.
It is known that Gibbs potential must be the minimum at the equilibrium under the constant pressure and temperature constraint. It makes me guess that Gibbs potential is a convex function of P and T.
However, mathematically, G must be a concave function of P and T. (It would be better if you are able to see Problem 1.10 of Plischke's book.) For example, 
However, it makes me guess that Gibbs potential has the maximum value at the equilibirum, but I know it is not true.
How can I understand about the concavity of Gibbs potential?
 A: 
It is known that Gibbs potential must be the minimum at the equilibrium under the constant pressure and temperature constraint. It makes me guess that Gibbs potential is a convex function of P and T.

This is the weak point. There is not such an implication.
The fundamental reason is that the minimum principle refers to a minimum with respect to variables expressing any possible constraint one could add to an equilibrium system at constant thermodynamic macrostate variables. It is not about a minimum of the thermodynamic potential with respect to its variables. In other words, the connection between the minimum principle for a thermodynamic potential and the sign of its second derivatives is not a straightforward application of the condition on the second derivatives at a point of minimum. It is more indirect. This is evident once we realize that the minimum principle holds for all the Legendre transforms of a thermodynamic potential, even if the definition of Legendre transforms used in Thermodynamics implies a convex/concave alternation.
The correct chain of implications is
minimum principle $\color{orange}\Rightarrow$ strict convexity with respect to the extensive variables in all the unique equilibrium states (i.e., in the absence of phase coexistence) $\color{orange}\Rightarrow$ the thermodynamic potential depending only on extensive variables (i.e., the internal energy) is a strictly convex function of all its variables $\color{orange}\Rightarrow$  every Legendre transform of the internal energy with respect to one extensive variable is a strictly concave function of the conjugate variable of that extensive variable.
I hope this clarifies why the concavity of the Gibbs free energy with respect to $P$ and $T$ does not contradict the minimum principle.
