# Max-min inequality

It is known that $\underset{x}{\max} \underset{y}{\min} f(x,y) \leq \underset{y}{\min} \underset{x}{\max} f(x,y)$ . When does equality hold in this expression?

Let $\max_x \min_y f(x, y) = \max_x g(x) = g(a)$ and $\min_y \max_x f(x, y) = \min_y h(y) = h(b)$. We have $$\max_x \min_y f(x, y)= g(a) \le f(a, b) \le h(b) = \min_y \max_x f(x, y)$$
So for equality, we need an element $(a, b)$ to exist s.t. $\min_y f(a, y) = f(a, b) = \max_x f(x, b)$.
Equality holds if $f(w,z)$ is convex in w and concave in $z$ (or convex in $z$ and concave in $w$)