In the field of convex optimization, and in particular on prof. Stephen Boyd's lecture slides, the conjugate $f^*$ of a function $f$ is defined to be
$$ f^*(y) = \sup_{x \in domf} (y^Tx - f(x))$$
I'm just trying to better understand this definition (what does it mean exactly? What is the geometrical interpretation?) and figure out how this is actually used in concrete. For sure, an interesting point is that $f^*$ is convex even if $f$ is not.
Many thanks,
James