Convexity under diffeomorphisms Let $K \subset \mathbb{R}^n$ be a compact convex subset with non-empty interior, and $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ a diffeomorphism. Then is it true that $f[K]$ is convex?

I'm afraid that in general it's not true. In $\mathbb{R}^2$ you can construct a diffeomorphism $(x, y) \mapsto (x + hy^2, y), \; h > 0$ and apply it to unit square (or unit circle). The "top part" of this figure will be stretched much stronger than parts are near to the $Ox$ axis, which will lead to fail of convexity. 
ADDITION: this is an illustration of what happens with rectangle under the action of this map (sorry for the quality, my usual drawing tool is on vacation :) )

