Editing Convex conjugate (section)

=== Order reversing===

Declare that <math>f \le g</math> if and only if <math>f(x) \le g(x)</math> for all <math>x.</math> Then convex-conjugation is [[Order theory|order-reversing]], which by definition means that if <math>f \le g</math> then <math>f^* \ge g^*.</math> 

For a family of functions <math>\left(f_\alpha\right)_\alpha</math> it follows from the fact that supremums may be interchanged that

:<math>\left(\inf_\alpha f_\alpha\right)^*(x^*) = \sup_\alpha f_\alpha^*(x^*),</math>

and from the [[max–min inequality]] that 

:<math>\left(\sup_\alpha f_\alpha\right)^*(x^*) \le \inf_\alpha f_\alpha^*(x^*).</math>