Which n-Venn diagrams can be drawn with convex k-gons?
<|reference_start|>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 >= (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].<|reference_end|>
@article{carroll2005which,
    title={Which n-Venn diagrams can be drawn with convex k-gons?},
    author={Jeremy Carroll (1), Frank Ruskey (2), Mark Weston (2) ((1) HP
  Laboratories, Bristol, UK, (2) University of Victoria, Canada)},
    journal={arXiv preprint arXiv:cs/0512001},
    year={2005},
    archivePrefix={arXiv},
    eprint={cs/0512001},
    primaryClass={cs.CG}
}