Exercise 3, § 2

Let A and B be two disjoint convex sets in a vector space E. Show that there exist two disjoint convex sets C, D in E such that $A \subset C, B \subset D$ and $C \cup D = E$. (Apply th. 2 of S, III, § 2.4 to the set of pairs of disjoint convex sets (M, N) such that $A \subset M$ and $B \subset N$ and express the fact that M and N do not meet by the relation $0 \notin M - N$. To show that $C \cup D = E$, obtain a contradiction supposing that $x_0 \notin C \cup D$; if $C'$ (resp. $D'$) is the convex envelope of $C \cup \{ x_0 \}$ (resp. $D \cup \{ x_0 \}$), show that it is impossible that both $C' \cap D \neq \varnothing$ and $C \cap D' \neq \varnothing$.)