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

TL;DR

This paper establishes a new lower bound on the number of sides convex polygons must have to form simple n-Venn diagrams, showing that certain configurations are impossible with triangles and providing examples with quadrilaterals.

Contribution

It introduces a new lower bound for sides of convex polygons in Venn diagrams and constructs an example achieving this bound with quadrilaterals.

Findings

Convex k-gons require at least (2^n - 2 - n) / (n (n-2)) sides for simple n-Venn diagrams.

7-curve Venn diagrams cannot be formed from triangles.

A simple, symmetric 7-Venn diagram with quadrilaterals is constructed.

Abstract

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].