The minimax theorem states the following:
Let $X\subset \mathbb{R}^{n}$ and $Y\subset \mathbb {R} ^{m}$ be
compact convex sets. If $ f:X\times Y\rightarrow \mathbb {R} $ is a
continuous function that is convex-concave, i.e.
$$f(\cdot ,y):X\rightarrow \mathbb {R} \text{ is convex for fixed }
y, \text{and}$$ $$ f(x,\cdot ):Y\rightarrow \mathbb {R} \text{ is concave for fixed } x.$$
Then we have that
$$ \min _{x\in X}\max _{y\in Y}f(x,y)=\max _{y\in Y}\min _{x\in
> X}f(x,y).$$
For arbitrary functions $f$ the equality does not hold in general. However, the Max-Min-Inequality is always satisfied.
For any function $f: Z \times W \to \mathbb{R}$ we have
$$ \inf _{w\in W}\sup _{z\in Z}f(z,w) \geq \sup _{z\in Z}\inf_{w\in W}f(z,w) .$$
Since this property always holds for arbitrary functions $f$, it is well worth to be kept in mind by students. Naturally, many students will tend to confuse the order of the max and min operations as well as the direction of the inequality. In a lecture on nonlinear optimization our professor told us the following mnemonic to remember the property, which, as I think, makes it really easy to remember:
The shortest giant is at least as tall as the tallest dwarf.
Here, the shortest giant refers to the inf sup on the left hand side, which is at least as tall (i.e. $\geq$) as the tallest dwarf, which corresponds to sup inf.
I haven't forgotten the Max-Min-Inequality since I learned the above mnemonic which is why I posted the question whether you are aware of other such neat mnemonics which make students' learning easier.
