Proposition 2

Let $E$ and $F$ be two affine spaces and $f$ be an affine linear mapping of $E$ in $F$; then the image of a convex subset of $E$ under $f$, and the inverse image of a convex subset of $F$ under $f$ are both convex.

The image under $f$ of the closed segment with end points $x, y$ is the closed segment with end points $f(x), f(y)$, hence the first statement. We deduce that the inverse image of a closed segment of $F$ under $f$ contains each closed segment whose end points belong to it; the second statement of prop. 2 follows.

In particular the image of a convex set under a homothety or a translation is a convex set.