Functions where the pre-image of convex sets is convex For functions $f:\mathbb R\to\mathbb R$, I've noticed an interesting property:
$f$ is monotonous exactly if the pre-images of convex sets are convex.
Now the latter condition can of course be defined for any map between real linear spaces. Obviously there it won't correspond to monotony (after all, what would it mean that a function $\mathbb R^m\to\mathbb R^n$ is monotonous?).
However I wonder: Is there any other intuitive property that is connected to the demand that pre-images of convex sets are convex?

This is rather a comment.
Actually, your observation in the scalar case is equivalent to "the monotone functions are exactly the quasilinear functions", see https://en.wikipedia.org/wiki/Quasiconvex_function.
Further, it is not hard to see that


*

*For all convex $C \subset \mathbb{R}^n$, the preimage $F^{-1}(C)$ is convex

*For all $x,y \in \mathbb{R}^m$, $\lambda \in [0,1]$, the point $F(\lambda \, x + (1-\lambda) \, y)$ belongs to the convex hull of $F(x)$ and $F(y)$.


are equivalent. Note that the second bullet implies that subspaces are mapped to subspaces.
Finally, I would like to point out that there is already a notion of "monotone" (it generalizes monotonically increasing functions) for functions mapping $\mathbb{R}^n$ to $\mathbb{R}^n$ (or, more generally, a Banach space $X$ into its topological dual space $X^*$), see https://en.wikipedia.org/wiki/Monotonic_function#Monotonicity_in_functional_analysis.
