Vapnik-Chervonenkis classes with the maximum property are in some sense the most perfect set systems of finite Vapnik-Chervonenkis dimension. Definability of maximum VC classes in model theoretic structures is closely tied to other measures of complexity such as dp-rank. In this talk we show that set systems realizable as sets of positivity for linear combinations of real analytic functions have the maximum property on sets in general position. This may have applications to proving lower bounds on dp-rank in certain theories.