Intersecting Unions of Maximal Convex Sets
Marilyn Breen · Proceedings of the American Mathematical Society · 1973
Hare and Kenelly have characterized the intersection of the maximal starshaped subsets of a set $S$, where $S$ is compact, simply connected and planar, and Sparks has solved the general problem for maximal ${L_n}$ sets. In this paper, a similar question is examined for unions of maximal convex sets: Let $S$ be a subset of ${R^2},\mathcal {C}$ the collection of all maximal convex subsets of $S$, and $\mathcal {N} = \{ A \cup B:A,B$ distinct members of $\mathcal {C}\}$. Then ${ \cap ^\mathcal {N}}$ is expressible as a union of three or fewer convex sets.