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

*Jeremey Carroll*,
HP Labs,
Bristol, England.

*Frank Ruskey*,
Department of Computer Science,
University of Victoria, Canada.

*Mark Weston*,
Department of Computer Science,
University of Victoria, Canada.

### 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
Grünbaum had constructed a 7-Venn diagram of non-convex 5-gons
["Venn Diagrams II", *Geombinatorics* 2:25-31, 1992].

Back to list of publications.