I know that the conditions to have that
$\min_{y\in Y} \max_{x\in X} f(x,y)=\max_{x\in X} \min_{y\in Y} f(x,y)$
are that $X,Y$ are compact and convex subsets of the product of reals and that $f$ is continuous and quasi-concave on $y$ and quasi-convex on $x$. I am trying to find particular cases in which I violate one of the conditions and get an inequality.
In particular, I am struggling to find $X$ or $Y$ not convex such that for $f: X\times Y \rightarrow R$, continuous, quasi-concave on $y$ and quasi-convex on $x$,
$\min_{y\in Y} \max_{x\in X} f(x,y)\neq \max_{x\in X} \min_{y\in Y} f(x,y)$.
I think my issue is probably related to a misunderstanding of the theorem (I can prove it but cannot check its conclusion).
Take a set $X$ with just two points. On $X\times X$ define $f(x,y)=1$ if $x=y$ and $0$ otherwise. This is example where min and max cannot be switched but $f$ is continuous.