If we know the convex conjugate of $f(x,y)$, what can we say about the conjugate of $f$ in $x$?

Question

Say $f^*(x,y)$ is the convex conjugate of $f(x,y)$. Now take $g_{y_0}(x) := f(x, y_0)$. Is there any relationship between $g^*_{y_0}(x)$ and $f^*(x, y_0)$?

Answer 1

They are of course the same. Writing $g_{y_0}(x)$ instead of $f(x,y_0)$ is only a typographical device to make it more evident that you consider $f$ as a function of $x$ alone for the moment.