Consider a function $f : \mathbb{R^d} \to \mathbb{R}^d$, with $d\geq 2$, such that:

• $f$ is injective,
• For any convex set $A$ of $\mathbb{R}^d$, $f(A)$ is also convex.

What can we say about $f$ ? In particular, is $f$ necessarily affine ? I tend to think yes, but I can't prove it.

Consider a function $f : \mathbb{R}^d \to \mathbb{R}^d$, with $d\geq 2$, such that:

• $f$ is injective,
• For any convex set $A$ of $\mathbb{R}^d$, $f(A)$ is also convex.

What can we say about $f$ ? In particular, is $f$ necessarily affine ? I tend to think yes, but I can't prove it.