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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.

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See Meyer, Walter; Kay, David C., A convexity structure admits but one real linearization of dimension greater than one, J. Lond. Math. Soc., II. Ser. 7, 124-130 (1973). ZBL0271.52002.

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.

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    $\begingroup$ Having just recently explained to my students the distinction between linear and affine, I am somewhat amused that the authors chose not to use the word "affine". $\endgroup$
    – Willie Wong
    Commented Nov 22, 2021 at 20:23

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