Given a function , we have several ways of checking if it is convex. But how to recognize the functions that are *convex up to a reparametrization*, meaning that there is a homeomorphism of the real line such that the composition is convex?

Let’s start by listing some necessary conditions for to have this property.

1. Since convex functions are continuous, must be continuous.

2. Convex functions are *quasiconvex*, meaning that each sublevel set is convex (in this case, an open interval). This property is preserved under reparametrization. So, must be quasiconvex as well.

3. Nonconstant convex functions are unbounded above. So, must have this property too.

Is this all? No. For example, let if and if . This is a continuous quasiconvex function that is unbounded above. But it can’t be made convex by a reparametrization, because as , it increases while staying bounded. This leads to another condition:

4. If is not monotone, then its limits as and must both be .

And still we are not done. The function

satisfies all of 1–4. But it has a flat spot (the interval where it is constant) which does not achieve its minimum. This can’t happen to convex functions: if a convex function is constant on an interval, then (by virtue of being above its tangent lines) it has to attain its minimum there. This leads to the 5th condition:

5. is either strictly monotone, or attains its minimum on some closed interval (possibly unbounded or a single point). In the latter case, it is strictly monotone on each component of .

Finally, we have enough: together, 1–5 is the necessary and sufficient condition for to be convex up to a reparametrization. The proof of sufficiency turns out to be easy. A continuous strictly monotone function is a homeomorphism between its domain and range. The strictly monotone (and unbounded) parts of give us homeomorphisms such that the composition of with them is affine. Consequently, there is a homeomorphism of the real line such that is one of the following five functions:

- if
- if is unbounded, but not all of
- if is a bounded nondegenerate interval
- if is a single point
- if is empty

Incidentally, we also see there are five equivalence classes of convex functions on , the equivalence being any homeomorphic change of variables.

What functions on are convex up to reparametrization? Unclear. (Related Math SE question)

You forget the monotone case which reparametrizes to $g(x)=x$ in your list of possibilities.

Thanks for pointing this out. I added the 5th case.