What automorphisms of $R^n$ preserve convexity? I believe I'm right in saying that an isometry of a convex figure will be convex. There are other automorphisms that have this property, for example "stretching along one axis" (e.g. $f(x,y) = (2x,y)$) won't transform a convex shape into a non-convex one or vice versa.
What other maps have this property?
 A: Search google books for "fundamental theorem of geometry" (it's in a book by M. Berger).
Using the fact that the only field automorphism of $\mathbb{R}$ is the identity, you get your result (for $n \geq 2$): the affine bijections.
A: Any transformation that sends lines to lines will do, since convexity is defined by the existence of certain lines in a figure.  In particular, any transformation of the form $x \mapsto Ax + b$ where $A$ is a matrix and $b$ is a vector works.  (It's not clear to me what you mean by "automorphism" here - in what category? - so depending on your meaning these are the only ones, by which I mean that as an affine space these are the only automorphisms of $\mathbb{R}^n$.)
If you don't trust that explanation, one way to see this is to prove the result for translations (obvious) and then prove it for elementary matrices.  It should be geometrically obvious for all of them, as long as you have a reasonably good handle on shearing.
Edit: In fact, I think this condition is necessary.  If a line $L$ is not sent to a line, draw a circle which is split in half by $L$.  Then the image of either one half or the other will not be convex.  
