Properties of functions having the form $g(x,t) = t f(\frac{x}{t})$

Question

I have been frequently coming across the function $g(x,t) = t f(\frac{x}{t})$ in my course on convex optimization. A friend of mine mentioned that it is the perspective function, but the book on convex optimization by Stephen Boyd gives a different definition of the perspective function.

This function also has the property of being convex in both $x$ and $t$ if $f(x)$ is convex. Does this function have any specific name so that I can look up its properties?

Answer 1

There's a subtle difference in nomenclature. What you've defined is what Boyd and Vandenberghe call the perspective of a function (c.f. Section 3.2.6), not the perspective function/mapping (c.f. Section 2.3.3). As they state, the concepts are related:

Therefore $\text{epi } g$ is the inverse image of $\text{epi } f$ under the perspective mapping that takes $(u, v,w)$ to $(u,w)/v$.

In other words, the epigraph of the perspective of a function is the inverse image of the epigraph of the function under the perspective mapping.