Proposition [lcc-001Q]

Let f: V \to \mathbb {R} be convex, so that (V,*,f) is a minmax problem. Then we can form the modified minmax problem L = (V,V^*,(x,\alpha ) \mapsto f(x) - \alpha (x)) - note that, up to a sign change in the domain, this amounts to adding the constraint x = 0.

Then (L^*)^+ = -L^- = f^*

Note that the two uses of the asterisk in this equation conflict. We have both the reversed optimization problem L^* given by flipping the variables, and the convex conjugate function f^*. It may be good to alter this notation to resolve the conflict.