Additional assumption to have a convex image Let $f\colon \mathbf{R}^2 \to \mathbf{R}^2$ a continuous injective function.
In general, it is not true that the image $f[X]$ is convex whenever $X \subseteq \mathbf{R}^2$ is convex.
Is there some additional assumption to ensure that $f[X]$ is convex? 

I'm not sure if this is the level of answer that you're looking for, but there is a whole calculus of convexity-preserving operations. Stephen Boyd has some lecture slides here:
http://stanford.edu/class/ee364a/lectures/functions.pdf
In particular, affine functions are convexity preserving. 
Other good sources about this would be Boyd's book "Convex Optimization" or the book "Fundamentals of Convex Analysis" by Hiriart-Urrut and Lemaréchal.
