Generalized inverses of strictly monotone transformations

Let $ C \subseteq \mathbb{R}^n $ be a convex set. The mapping $ f : C \longrightarrow \mathbb{R}^n $ is strictly increasing, if $ \langle f(x)-f(y), x-y \rangle > 0 $ for all distinct elements $ x,y \in C $. Applying classical theorems of finite dimensional convex geometry and convex analysis, we show that the inverse functions has a unique extension $ f^{(-1)} : \mathrm{conv} (f(C)) \longrightarrow C $ such that $ f^{(-1)} $ is monotone, continuous and it acts as a left-inverse of $ f $. As an application we introduce the concept of vector-valued weighted quasi-arithmetic means and discuss their equality problem.