Functions that are convex after monotonic change of variables

Question

Consider a multivariate function $y = f(x_1,...,x_n)$. Suppose there exist monotonic functions $h_i, g$, such that the function:

$$F(y_1,...,y_n) = g(f(h_1(y_1), ..., h_n(y_n)))$$

is convex.

Is there a name for functions $f$ satisfying this property? In this sense they are "almost convex".

Answer 1

I think the functions you are describing are quasiconvex.

For univariate functions, these are equivalent classes, I think. For multivariate functions, it's less clear to me. Maybe instead of monotonic functions in each variable, you should replace with cyclical monotonicity?

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NOTE: Based on comments by Calvin below, it should be required that $h$ and $g$ be homeomorphisms (I think continuity + strict monotonicity + surjectivity suffice in $\mathbb{R}$), for equivalence to quasiconvexity in the 1D case. See comments for counterexamples otherwise.