What is a convex conjugate ?

In case you've missed the convex-optimization revolution (which started with Mureau, Rockafellar, etc., but has pre-history dating back to at least Kuhn, Karush, Tucker, Nash, etc.), the hottest things around are subdifferentials, convex conjugates (aka Fenchel-Legendre transforms), prox operators, and infimal convolutions. Checkout this beautiful one-page manuscript on the subject, by H.H. Bauschke. There are many cool stuff associated with this duality result, here are a few:

• All primal solutions are obtainable from any dual solution! Paramount is the Fenchel duality Theorem (see previous reference), a monster result for mapping between an arbitrary dual solution and the entire set of primal solutions. You can see it in action here.
• Duality between strong convexity and strong smoothness. Another paper shows a non-trivial and very useful result, Theorem 6 (it's fairly possibly this result was known earlier, but this is another debate...): a closed convex function $R$ is $\beta$-strongly convex w.r.t to a norm $\|.\|$ iff its Fenchel-Legendre transform (aka convex conjugate) $R^*$ is $\frac{1}{\beta}$-strongly smooth w.r.t the dual norm $\|.\|_*$. This result can be invoked to establish the strong convexity of some otherwise rather complicated matrix functions.