I am bad at math and am having an extremely hard time trying to understand the conjugate of a function. I am not sure why the dotted line is the conjugate function. How do I find that? This is my limited understanding of the steps.
1) draw $y^Tx$ which is equals to $x y$ in 2 dimensions. Question: is $x$ my gradient or is $y$ my gradient? Because the original equation is $f(x)$ but now is $f^* (y)$?
2) Maximize the function $x y - f(x)$ with respect to $f$ because I am to find the supremum of the expression. How do I draw this?
So I only know how to draw the line $x y$. Then I am stuck. It would be nice if someone could give a numerical example by stepping through some x or y values ( i am not sure if i am supposed to step through the x or y values). I can only understand after seeing an example with numbers.
Question: Why do I shift the line downwards (blue gap) ? The red gap looks bigger to me and we are finding the maximum.