In this paper, we study the minimization of the max function of q smooth convex functions on a domain specified by infinitely many linear constraints. The difficulty of such problems arises from the kinks of the max function and it is often suggested that, by imposing certain regularization functions, nondifferentiability will be overcome. We find that the entropic regularization introduced by Li and Fang is closely related to recently developed path-following interior-point methods. Based on their results, we create an interior trajectory in the feasible domain and propose a path-following algorithm with a convergence proof. Our intention here is to show a nice combination of minmax problems, semi-infinite programming, and interior-point methods. Hopefully, this will lead to new applications.