Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be compact sets.
Consider a continuous function $f : X \times Y \rightarrow \mathbb{R}$.
Say under which condition we have
$ \min_{x \in X} \max_{y \in Y} f(x,y) = \max_{y \in Y} \min_{x \in X} f(x,y). $
From this we have that $\max_{y \in Y} \min_{x \in X} f(x,y) \leq \min_{x \in X} \max_{y \in Y} f(x,y)$. So here we are looking for conditions on $f$ such that we have the equality.