Which n-Venn diagrams can be drawn with convex k-gons?

Details Which n-Venn diagrams can be drawn with convex k-gons? Which n-Venn diagrams can be drawn with convex k-gons?

2005-11-30

arxiv.org/abs/cs/0512001v1

Computer Science

Computational Geometry

10 pages, 3 figures. To be published in Proceedings of the Second International Workshop on Euler Diagrams (Euler 2005), Elect

Scientific paper

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].

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