  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).
