I am trying to understand the pointwise supremum from Boyd & Vandenberghe's Convex Optimization.
Let me try to share my understanding, please validate it. Let's try to visualise $g(x)$. $f(x,y)$ is a two variable function. for any particular x value say X and all y values we select maximum of $f(X,y)$ which is the slice of the surface $f(x,y)$ where y is varying over the entire y axis for a particular X?
What does the following mean?
$ y \in \mathcal A $
What is the set $ \mathcal A $? is it the set of y defined for function $f(x,y)$ and not the entire y axis?
the resulting g(x) may not be continuous right? but it maybe be convex?
Are $f(X,y)$ slices the infinite set of convex functions? Is it because $x$ and $y$ are on the reals?