Let $X$ be a topological space, $Z$ be a compact convex subset of $\mathbb R^m$, and let $f:X \times Z \to \mathbb R$ be a continuous function (w.r.t the product topology on $X \times Z$).
Question. Under what minimal additional assumptions do we have the duality $\sup_{z \in Z}\inf_{x \in X}f(x,z) = \inf_{x \in X}\sup_{z \in Z}f(x,z)$ ?
The case $m=1$ (in which case $Z$ must be a compact interval in $\mathbb R$) was proved in Theorem 1 of Minimax theorems in a fully non-convex setting under additional connectivity conditions on the sets $\arg\min f(\cdot,z)$ and $\arg\max f(x,\cdot)$ for all $(x,z) \in X \times Z$.
