TITLE: Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?
QUESTION [14 upvotes]: Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets.
Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$.
Can $\text{Proj}(P)$ have more than $f$ facets?
In the general case, each successive projection can increase the number of facets from $f$ to $\left\lfloor \frac{f^2}{2} \right\rfloor$, but I'm wondering if $\mathbb{R}^2$ is a special case.

REPLY [23 votes]: Your question is essentially about extension complexity.  In general, the extension complexity of a polytope $P$ is the minimum number of facets over all polytopes $Q$ which project to $P$.  You are interested in the extension complexity of polygons.  Fiorini, Rothvoß, and Tiwary proved that regular $n$-gons have extension complexity $O(\log n)$.  For lower bounds, they give examples of $n$-gons which have extension complexity $\sqrt{2n}$.
It is an open question if there exists for infinitely many $n$, an $n$-gon with extension complexity $\Omega(n)$.
Edit. Shitov has recently shown that every convex $n$-gon has extension complexity at most $147 n^{2/3}$, which resolves the open question in the negative.
As a bonus, here is a picture of the polytope in Nate's answer (courtesy of Samuel Fiorini).