Maurice Sion's paper "On General Minimax Theorems" aims to unify and extend the minimax theorems for functions that are quasi-concave-convex and semi-continuous in each variable. The paper begins by reviewing von Neumann's minimax theorem and its various generalizations, including those by J. Ville, A. Wald, M. Shiffman, H. Kneser, K. Fan, and C. Berge. These generalizations often rely on the separation of disjoint convex sets or fixed point theorems, but they do not cover all cases, especially for quasi-concave-convex functions. Sion introduces the concepts of concave-like, convex-like, quasi-concave, and quasi-convex functions, which generalize the traditional notions of concavity and convexity. He then presents a minimax theorem for quasi-concave-convex functions using a novel proof method based on a theorem by Knaster, Kuratowski, and Mazurkiewicz, which is derived from Sperner's lemma. This method avoids the use of fixed point theorems and hyperplane separation due to the lack of continuity and convexity. The paper includes several lemmas and theorems that support the main result. Theorem 3.4 states that if \( M \) and \( N \) are convex, compact spaces, and \( f \) is a quasi-concave-convex function on \( M \times N \) that is upper semi-continuous in \( M \) and lower semi-continuous in \( N \), then \( \sup \inf f = \inf \sup f \). This theorem is proven using a constructive approach involving finite sets and the properties of quasi-concave-convex functions. Corollary 3.6 demonstrates that the condition of upper semi-continuity in \( M \) cannot be relaxed without affecting the minimax equality. The paper also discusses minimax theorems for concave-convexlike functions, extending Kneser's and Fan's results to more general spaces. Overall, Sion's work provides a comprehensive and unified framework for understanding minimax theorems, particularly for functions that are quasi-concave-convex and semi-continuous.Maurice Sion's paper "On General Minimax Theorems" aims to unify and extend the minimax theorems for functions that are quasi-concave-convex and semi-continuous in each variable. The paper begins by reviewing von Neumann's minimax theorem and its various generalizations, including those by J. Ville, A. Wald, M. Shiffman, H. Kneser, K. Fan, and C. Berge. These generalizations often rely on the separation of disjoint convex sets or fixed point theorems, but they do not cover all cases, especially for quasi-concave-convex functions. Sion introduces the concepts of concave-like, convex-like, quasi-concave, and quasi-convex functions, which generalize the traditional notions of concavity and convexity. He then presents a minimax theorem for quasi-concave-convex functions using a novel proof method based on a theorem by Knaster, Kuratowski, and Mazurkiewicz, which is derived from Sperner's lemma. This method avoids the use of fixed point theorems and hyperplane separation due to the lack of continuity and convexity. The paper includes several lemmas and theorems that support the main result. Theorem 3.4 states that if \( M \) and \( N \) are convex, compact spaces, and \( f \) is a quasi-concave-convex function on \( M \times N \) that is upper semi-continuous in \( M \) and lower semi-continuous in \( N \), then \( \sup \inf f = \inf \sup f \). This theorem is proven using a constructive approach involving finite sets and the properties of quasi-concave-convex functions. Corollary 3.6 demonstrates that the condition of upper semi-continuity in \( M \) cannot be relaxed without affecting the minimax equality. The paper also discusses minimax theorems for concave-convexlike functions, extending Kneser's and Fan's results to more general spaces. Overall, Sion's work provides a comprehensive and unified framework for understanding minimax theorems, particularly for functions that are quasi-concave-convex and semi-continuous.