We have two functions, $f$ and $g$ which are convex, differentiable everywhere, non-increasing and defined on the unit square. The graphs have a unique intersection point $(x, y)$ of interest. Now assume we take the convex conjugates of $f$ and $g$. In the dual space of the convex conjugate, what point does $(x,y)$ become mapped to?
In essence I am asking the inverse question of "if the convex conjugates intersect, what happens to the primal functions?" to which the answer is "the point of intersection becomes the unique tangent to both primal functions". My conjecture is that it's the same, i.e. the point where the convex conjugates have the same slope, but I was not able to confirm it so far.