I was reading the book convex optimization by Boyd and Vandenberghe,while reading the chapter about convex functions,there is a part about operations that preserves convexity and the perspective of a function is one of those operations,according to the book:
I understand the definition and the proof why it preserves convexity,but in the earlier chapter there is a perspective function
I find these two functions very similar and i like the intuitive it gives about the perspective function,so I just wonder is there an intuitive understanding of the the perspective of a function and is it a generalized form of the perspective function?
Best Answer
If the perspective function $P$ is like the action of a pinhole camera, then the perspective of a function $f$ is the function $g$ whose graph when viewed through a pinhole camera looks like the graph of $f$. This is like the action of a projector, or a laser light show.
Briefly, $(\mathbf x,t,y)$ is in the graph of $g$ if and only if $P(\mathbf x,y,t) = (\mathbf x,y)/t$ is in the graph of $f$. (Note that the coordinates have been rearranged slightly to make sure that perspective is applied along the $t$ axis.)
All this makes more sense if you first define the perspective of a set $\mathcal S\subseteq\mathbb R^n$ as the set of points in $\mathbb R^{n+1}$ which the perspective function maps into $\mathcal S$, show that this operation preserves convexity of sets, and apply it to the epigraph of $f$. This is left as an exercise for the reader.