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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 ≥ (2n - 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 convex quadrilaterals. Previously Grunbaum had constructed a symmetric 7-Venn diagram of non-convex 5-gons ["Venn Diagrams II", Geombinatorics 2:25-31, 1992].

Which n-Venn Diagrams Can Be Drawn with Convex k-Gons?