Let $f: R^2 \to (-\infty;+\infty]$
a function of two variables $(x,y)$.
Let $f^*(u,v) = \sup_{x,y}(xu + yv)$ be its Fenchel transform.
Find the Fenchel transform $f(x,y_0)$ when fixing $y=y_0$ as for a function of one variable $x$.
There is an idea to use the formula for $(f + g)^*$ and divide the original function of two variables into auxiliary functions $f_1 (x, y_0)$ and $f_2 (x_0,y)$ with one fixed variable, but I haven’t figured out how to proceed further and bring the problem to response.