On amount of minimal pairs (Q1902325)

Let \({\mathcal K} (X)\) denote the set of all compact convex sets in a finite dimensional linear space \(X\). For a given set \(F \in {\mathcal K} (X)\), denote by \({\mathcal M} (F)\) the set of all pairs \((A,B)\), \(A,B \in {\mathcal K} (X)\), such that \(A + B = F\). The minimality in \({\mathcal M} (X)\) is considered with respect to the relation: \((A,B) \leq (C,D)\) if and only if \(A \subset C\), \(B \subset D\). It is proved that the set of all minimal pairs, as well as the set of all non-minimal pairs, are both dense in \({\mathcal M} (F)\).