\(\Phi-\)convexity, a form of abstract convexity, was first introduced by Ky Fan [2] and next
investigated by Pallaschke and Rolewicz [3], Rubinov [4],
Singer [6] and many other authors.
The present talk is devoted to minimax theorems for \(\Phi-\)convex functions.
Starting from the paper by Ky Fan [1]
convexlike properties were used in those minimax theorems
which do not refer to linear structures of the underlying spaces.
Let \(X\) be a set and \(\Phi\) be a class of functions \(\varphi:X\rightarrow \text{R}
\).
Following Ky Fan [1] we say that the class \(\Phi\)
is {\em convexlike on \(X\)}
if for any \(x_{1},x_{2}\in X\) and \(t\in[0,1]\) there exists \(x_{0}\in X\) such that
\[
\varphi(x_{0})\le t\varphi(x_{1})+(1-t)\varphi(x_{2})\ \ \ \text{for}\ \ \ \varphi\in \Phi.
\]
Numerous extensions or generalizations of convexlikeness
have been proposed (see for example [1], [5]).
We introduce joint convexlikeness
which generalizes the convexlikeness
and is shaped for \(\Phi\)-convex functions.
The property of joint \(\Phi-\)convexlikeness allows us to obtain minimax theorem for
functions with not necessarily connected
level sets.
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References- K. Fan, Minimax theorems , Proc. Nat. Acad. Sci., vol. 39 (1953), 42-47.
- K. Fan, On the Krein-Milman theorem , Convexity, Proceedings
of Symp. Pure Math., vol.7, American Mathematical Society, Providence (1963), 211-219.
- D. Pallaschke, S. Rolewicz,
Foundations of Mathematical Optimization , Kluwer Academic, Dordecht, 1997.
- A.M. Rubinov, Abstract Convexity and Global Optimization , Kluwer Academic, Dordrecht, 2000.
- A. Stefanescu, The minimax equality; sufficient and necessary conditions , Acta Math. Sinica, English Series, vol. 23 (2007), 677-684.
- I. Singer, Abstract Convex Analysis ,
Wiley-Interscience, New York 1997.
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