A minimax theorem for functions with possibly nonconnected intersections of sublevel sets (Q2581478)

The authors apply a selection theorem from [\textit{P. V. Semenov}, Math. Notes 54, No. 6, 1236--1240 (1993); translation from Mat. Zametki 54, No. 6, 74--81 (1993; Zbl 0853.46008)] for multivalued mappings with paraconvex values (rather than various versions of KKM-principle) to prove the following minimax theorem and a couple of its generalizations. Let \(f: X\times Y \rightarrow R\) be a real-valued function on the Cartesian product of two \(AR\) subcompacta \(X\) and \(Y\) of a Banach space and suppose that: (1) for each \(c \in R\) and each \(x_0 \in X\), the set \(\{y\in Y/: f(x_0, y) \leq c \}\) is \(\alpha\)-paraconvex compact; and (2) for each \(d \in R\) and each \(y_0 \in Y\), the set \{\(x \in X: f(x, y_0) \geq d\) \} is \(\alpha\)-paraconvex compact for a fixed \(\alpha: (0, \infty ) \rightarrow [0, 1).\) Then \(\max_X\{\min_Y\{f(x, y)\}\}= \min_Y\{\max_X\{f(x, y)\}\}\).