Polygons as sections of higher-dimensional polytopes

We show that every heptagon is a section of a $3$-polytope with $6$ vertices. This implies that every $n$-gon with $n\geq 7$ can be obtained as a section of a $(2+\lfloor\frac{n}{7}\rfloor)$-dimensional polytope with at most $\lceil\frac{6n}{7}\rceil$ vertices; and provides a geometric proof of the fact that every nonnegative $n\times m$ matrix of rank $3$ has nonnegative rank not larger than $\lceil\frac{6\min(n,m)}{7}\rceil$. This result has been independently proved, algebraically, by Shitov (J. Combin. Theory Ser. A 122, 2014).