I was wondering about the following related questions:
Is there an upper bound for the number $n$ of sets for which a two-dimensional Venn diagram can be constructed using convex shapes only? What if we also require that the shapes be identical (but not necessarily symmetrically arranged)? What about in three or more dimensions?
Wikipedia shows a 6-set Venn diagram constructed with triangles, and a 5-set one constructed with identical ellipses. Thus the answers to the first two questions must be at least 6 and 5, respectively.
