Exercise 6, § 2

Let E be an affine space of dimension $\geq 2$ and let $f$ be a bijection of E on itself; show that if the image under $f$ of every convex subset of E is also a convex set, then $f$ is an affine linear mapping (consider the inverse mapping of $f$ and note that a closed segment is the intersection of the convex sets which contain its extremities cf. A, II, § 9, exerc. 7).