Minimax theorems for interval spaces
For a real-valued function a on the product \(X\times Y\) of two nonvoid sets conditions are given which imply the ``minimax equality'' \[ (1)\quad \inf_{y\in Y}\sup_{x\in X}a(x,y)=\sup_{x\in X}\inf_{y\in Y}a(x,y). \] The standard proofs of such minimax theorems either use a separation theorem or some version of Brower's fixed point theorem. It is the contribution of \textit{Wu Wen-tsün} [Sci. Rec., N. Ser. 3, 229-233 (1959; Zbl 0115.383)] to have observed that the only property of convex sets which is actually needed in the proof of the minimax theorem is connectedness. The authors ``want to demonstrate that \textit{L. Stachó}'s concept of an interval space [Acta Sci. Math. 42, 157-164 (1980; Zbl 0436.49017)], which generalizes the notion of a convex set, is an adequate frame for the formulation of a fairly general minimax theorem''. A simplified version of the main result reads as follows: Let X be a compact topological space and Y an interval space. Suppose that for all \(\alpha\in {\mathbb{R}}\) the sets \(\cap^{n}_{i=1}\{x\in X:\) \(a(x,y_ i)\geq \alpha \}\), \(y_ 1,...,y_ n\in Y\), \(n\in {\mathbb{N}}\), are either empty or closed and connected, and the sets \(\{\) \(y\in Y:\) a(x,y)\(\leq \alpha \}\), \(x\in X\), are closed and convex. Then (1) holds. The proof relies only on elementary connectedness arguments. The main theorem includes as special cases well known minimax theorems of Wu Wen- tsün, Hoang Tuy, Sion, Brézis-Nirenberg-Stampacchia, Stachó, and others.
scientific article; zbMATH DE number 4128677
scientific article