A minimax theorem is a theorem which states that under certain conditions on $\mathcal{X}$, $\mathcal{Y}$ and $f$: $$ \inf_{x \in \mathcal{X}}{\sup_{y \in \mathcal{Y}}{f(x,y)}} = \sup_{y \in \mathcal{Y}}{\inf_{x \in \mathcal{X}}{f(x,y)}}$$ All minimax theorems rely strongly on convexity: the sets $\mathcal{X}$ and $\mathcal{Y}$ are usually required to be convex subsets of vector spaces, and $f$ to be convex-concave.

My question is: of course one cannot expect much without some convexity, but is it possible to get rid of the convexity of the domain in any way?

For example, consider the simplest minimax problem: $\mathcal{X}$ is a subset of $\mathbb{R}^n$, $\mathcal{Y}$ a convex subset of $\mathbb{R}^n$ and $f(x,y) = \langle x,y \rangle$ is the usual scalar product.

Are there any minimax results in this setting (with of course additional assumptions on $\mathcal{X}$ or $\mathcal{Y}$, but $\mathcal{X}$ should not be assumed to be convex) ?