How to read 'max min' and 'min max'?
I'm frankly not even sure whether the above equation is meant to involve a multiplication.


No, not at all. We have nesting here:

$\max_{X \in \mathcal{X}}\ ( \min_{x \in X} x)$

is the largest minimum of all $X$. Similarly -- but the other way around -- for $\min \max$. This is very similar to nested quantifiers $\forall$ and $\exists$.

Let, for instance, $\mathcal{X} = \{ \{1,2\}, \{3,4\} \}$. Then, the following equalities hold:


$\min_{X \in \mathcal{X}} \min_{x \in X} x = 1$.
$\min_{X \in \mathcal{X}} \max_{x \in X} x = 2$.
$\max_{X \in \mathcal{X}} \min_{x \in X} x = 3$.
$\max_{X \in \mathcal{X}} \max_{x \in X} x = 4$.


Play around with the example by adding more sets to $\mathcal{X}$, more values to the individual sets, and quantify over $f(x)$ instead of just $x$ for different $f$, in order to get a feeling for this. 

If you are inclined to algorithmic thinking, do a quick implementation; you'll see how these operators nest more clearly with types, and examples are faster to compute. For instance, the second example would be

Xs.map { X =&gt; X.max }.min