Which mappings preserve convex bodies? Let $$f:\mathbb{R}^n\to\mathbb{R}^n,$$ $n\geq 2$, be a mapping which maps every convex body (compact convex set with nonempty interior) to a convex body. 
If we assume $f$ to be a homeomorphism, it needs to be affine. Is there something we can say without this assumption?
 A: Convexity is preserved under the class of Fractional Linear maps. They are of the form 
$$x\mapsto A(x)/f(x),$$ 
where $A:\mathbb{R}^n\to\mathbb{R}^n$ and $f:\mathbb{R}^n\to\mathbb{R}$ are affine maps. Note that these maps takes intervals to intervals, hence they preserve convexity.
Edit: You can see more on this in the paper "Order isomorphisms on convex functions in windows" by Artstein-Avidan, Florentin and Milman. In the paper the quote the classical theorem:
Theorem [The Fundamental Theorem of Affine Geometry]: Let $m≥2$ and $f:\mathbb{R}^n→\mathbb{R}^m$ be a bijective interval preserving map.  Then $f$ must be an affine transformation.
And prove,
Theorem: Let $n≥2$ and let $K⊆\mathbb{R}^n$ be a convex set with non empty interior.  If $F:K→R^n$ is an injective interval preserving map, then $F$ is a fractional linear map.
A: Let $h: {\mathbb R}^2\to [0,1]\subset {\mathbb R}$ be a continuous surjective map which is nonconstant on all nondegenerate intervals, for instance, we can take $h(x,y)=|\sin(x^2+y^2)|$. Let $g: [0,1]\to Q=[0,1]\times [0,1]\subset {\mathbb R}^2$ be a discontinuous function which is maps each nondegenerate interval onto the square $Q$, see the answer here. The composition $f=g\circ h$ then sends every nondegenerate interval onto the square $Q$. In particular, the image of each convex body under $f$ is again a convex body. The map $f$ will be discontinuous. It is unclear to me if there are continuous non-affine maps ${\mathbb R}^n\to {\mathbb R}^n$ which send convex bodies to convex bodies.  
P.S. The accepted answer does not produce valid examples since non-affine  linear-fractional (aka projective) transformations are not everywhere defined on the affine space. (They are well-defined on the projective space, of course.) 
