Generalized inverses of strictly monotone transformations

Let $C\subseteq {\mathbb{R}}^n$ be a convex set. The mapping $f:C⟶{\mathbb{R}}^n$ is strictly increasing, if $⟨f(x)-f(y),x-y⟩>0$ 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 $f^{(-1)}:\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(f(C))⟶C$ 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.