Question regarding convergence proof for Generative Adversarial Networks • /r/MachineLearning

While reading the Generative Adversarial Networks paper I encountered the following statement in section 4.2: "Proof. Consider V (G, D) U(pg, D) as a function of pg as done in the above criterion. Note that U(pg, D) is convex in pg." As far as I understand, pg is a probability distribution which in this case means that U(pg, D) is a function of many variables (the parameters of both pg and D combined). To say anything about the convexity of U(pg, D) we need to calculate the Hessian of U(pg, D), don't we?