  We establish a new lower bound for the number of sides required for the
component curves of simple Venn diagrams made from polygons. Specifically, for
any n-Venn diagram of convex k-gons, we prove that k >= (2^n - 2 - n) / (n
(n-2)). In the process we prove that Venn diagrams of seven curves, simple or
not, cannot be formed from triangles. We then give an example achieving the new
lower bound of a (simple, symmetric) Venn diagram of seven quadrilaterals.
Previously Grunbaum had constructed a 7-Venn diagram of non-convex 5-gons
[``Venn Diagrams II'', Geombinatorics 2:25-31, 1992].
