Max–min inequality

In mathematics, the max–min inequality is as follows: for any function f: Z × W → ℝ,

When equality holds one says that f, W and Z satisfies a strong max–min property (or a saddle-point property). As the function f(z,w)=sin(z+w) illustrates, this equality does not always hold. A theorem giving conditions on f, W and Z in order to guarantee the saddle point property is called a minimax theorem.

Define ${\displaystyle g(z)\triangleq \inf _{w\in W}f(z,w)}$.

${\displaystyle \Longrightarrow g(z)\leq f(z,w),\forall z,w}$

${\displaystyle \Longrightarrow \sup _{z}g(z)\leq \sup _{z}f(z,w),\forall w}$

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