Finite unions of convex sets

Abstract

In this paper it is shown that a set is the union of k convex subsets if and only if every finite subset of it is contained in some k convex subsets of it.This is a characterization of a set as the union of a finite number of convex sets by conditions on its finite subsets.Also, a proof of McKinney's theorem for unions of two convex sets is given using similar methods.Richard McKinney has given a characterization of unions of two convex sets.In this paper a complete characterization of unions of convex sets is given, and we give another proof of McKinney's result.Here, a k-partition of a set A is a family 6P={SX, S2, , Sk} of subsets of A, having k elements, where An5,J=0 for ijtj, and (JLi S=S.A property F of sets is said to be hereditary if, given any set A with property F, any subset F<= S has property P. I. First, we prove a theorem which enables us to by-pass further reference to Zorn's Lemma.