# Section of an $n$-dimensional convex polytope by $2$-dimensional plane

Consider an $n$-dimensional convex polytope with $k$ vertices. In the worst case the number of faces is exponential in $n$ and $k$. Consider a $2$-dimensional plane which intersects this polytope, i.e., it intersects only a subset of all faces. Can I bound the number of such intersected faces? In the worst case, will this number be a polynomial in $n$ and $k$ or still exponential?

• Also, you mean bound via $k$ for fixed $n$ or for $n$ varying with $k$? In the first case it is not true that the number of faces may be exponential. – Fedor Petrov Jul 21 '17 at 17:29
• For $n=2$ there are less than $2k$ faces. If you put a short pentagonal pyramid on each face of a dodecahedron you have $k=32$ and $60$ faces. That is probably the closest to $2$ that the ratio gets. – Aaron Meyerowitz Jul 22 '17 at 23:42
• For the purposes of visualization, it may be easier to consider the dual question. I haven't thought about this in detail, but perhaps it is this one: for an $n$-polytope with $k$ facets, what is the maximal number of vertices that a $2$-dimensional projection can have? – Tobias Fritz Jul 23 '17 at 21:40