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

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 Gr¨unbaum had constructed a symmetric 7-Venn diagram of non-convex 5-gons [“Venn Diagrams II”, Geombinatorics 2:25-31, 1992].

• Type: Journal
• Year: 2007
• Where: DCG
• Authors: Jeremy Carroll, Frank Ruskey, Mark Weston
• Keywords: DCG 2007, N-Venn Diagram, Simple Venn Diagrams, Venn Diagrams