Convexity without vector space structure (Q2707490)

\textbf{Stefan Nickel (Kaiserslautern)}: Let \(F\) be a family of real-valued functions on an arbitrary set \(X\) and \(t \in (0,1)\). \(F\) is \(t\)-convexlike on \(X\) if for any pair of points in \(X\) we can find a third one such that the usual convexity inequality is fulfilled for the chosen \(t\). \(F\) is convexlike on \(X\) if \(F\) is \(t\)-convexlike on \(X\) for every \(t \in (0,1)\). NEWLINENEWLINENEWLINEBased on these concepts Ky Fan proved in 1953 the first minimax theorem without algebraic structure of \(X\). The authors review in their paper these convexity concepts and several generalizations like weak-convexlike. In addition they propose some new convexlike-type properties which generalize the mean-convexity concept. NEWLINENEWLINENEWLINE\textbf{Rita Pini (Milano)}: The authors present a review and state the relationship among the concepts of \(t\)--convexlikeness, weak convexity, \(t\)--convexity, that are widely used in proving minimax results. These concepts apply to families of functions defined on an arbitrary nonempty set, and are essentially based on uniform convexlike properties of the functions of the family. In the end, the concepts of convexity, weak convexity and weak convexlikeness are generalized via a mean function.NEWLINENEWLINEFor the entire collection see [Zbl 0948.00038].