On reduced pairs of bounded closed convex sets. (Q1432791)

On the collection of ordered pairs of bounded closed convex sets in a real topological vector space define an equivalence relation \((A,B)\sim(C,D)\), meaning \(\overline{A+D}=\overline{B+C}\). On each equivalence class define a partial order \((A,B)\leq(C,D)\), meaning \((A,B)\sim(C,D)\) and \(A\subseteq C\), \(B\subseteq D\). A pair \((A,B)\) is {reduced} if for all \((C,D)\sim(A,B)\) there exists a bounded closed convex \(M\) such that \(C=\overline{A+M}\) and \(D=\overline{B+M}\). In particular, a reduced pair is a minimal element in its equivalence class. In this paper some properties of reduced pairs are proved.