Topological Minimax Theorems: Old and New

We review and extend the main topological minimax theorems based on connectedness that have been developed over the years since the pioneering paper of Wu (1959). It is shown in particular that the topological minimax theorems of Geraghty and Lin (1984) are essentially a rediscovery of much earlier results of Tuy (1974), while the latter can be derived from a minimax theorem recently developed for functions involving a real variable. Several new topological minimax theorems are presented, including a general theorem containing both König’s (1992) and Tuy’s (1974) results and a minimax theorem for increasing-decreasing functions analogous to Sion’s classical theorem. 2000 Mathematics Subject Classification. 49J35, 54D05.