# Functions of $\mathbb{R}^d$ preserving convexity of sets

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.

Theorem 4. If $$V$$ and $$W$$ are real vector spaces with $$\dim V>1$$, and $$f:V\to W$$ is a one-to-one mapping which preserves convexity, then $$f$$ is either linear (if $$f(0)=0$$) or is the translate of a linear map.