Let $P \subseteq \mathbb{R}^n$ be a polyhedron described by $\mathcal{O}(n^{c\_1})$ inequalities, where $c\_1$ is a constant. Moreover, let $M\colon P \to \mathbb{R}^2$ be a linear mapping. I'm looking for $P$ and $M$ such that the polyhedron $M(P)$ has $\mathcal{O}({c\_2}^n)$ vertices, where $c\_2$ is a constant.


Are there any other known examples besides [1](http://arxiv.org/abs/1308.2495), [2 (Theorem 4.4)](https://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCkQFjAA&url=http%3A%2F%2Fpage.mi.fu-berlin.de%2Fgmziegler%2Fftp%2Farchiv%2F051amentaziegler-deform.ps.gz&ei=gl68VPvdLs3JOYTSgYAI&usg=AFQjCNE-cRVNGaQWuwfx0KXweZvaOTkJ-w&sig2=GTlx1oNkwbqYCdQV7wT0BQ)?



Examples of Polyhedra with Large Shadows
If I understood your question right, you are looking for a polygon that has a small so-called *extended formulation*.
This problem is studied for example here:
<http://arxiv.org/abs/1107.0371>.


