Summary: We present new ideas and concepts in minimax equalities. Two important classes of multifunctions will be singled out, the weak Passy-Prisman multifunctions and multifunctions possessing the finite simplex property. To each class of multifunctions corresponds a class of functions. We obtain necessary and sufficient conditions for a multifunction to have the finite intersection property, and necessary and sufficient conditions for a function to be a minimax function. All our results specialize to sharp improvements of known theorems (Sion, Tuy, Passy-Prisman, Flåm-Greco). One feature of our approach is that no topology is required on the space of the maximization variable. In a previous paper [G. H. Greco and C. D. Horvath, Mathematika 47, No. 1-2, 9-18 (2002; Zbl 1012.52017)], we presented a “method of reconstruction of polytopes” from a given family of subsets, this in turn leads to a “principle of reconstruction of convex sets” (Theorem 3) which plays a major role in this paper. Our intersection theorems bear no obvious relationship to other results of the same kind, like KKM or other more elementary approaches based on connectedness. We conclude our work with a remark on the role of upper and lower semicontinuous regularization in minimax equalities.