On cutting frustum (Q1379863)

Let \(A\) and \(B\) be compact convex sets lying on parallel closed hyperplanes of a locally convex real topological vector space. The convex hull \(F\) of \(A\cup B\) is called the frustum over \(A\) and \(B\), and the intersection \(C\) of \(F\) with an intermediate parallel closed hyperplane is said to be a proper parallel cutting frustum. The author proves that the pair \((F_1, F_2)\) of frustums \(F_1\) over \(A\) and \(C\), and \(F_2\) over \(C\) and \(B\), is not minimal, that is, that there are compact convex proper subsets \(A_1\subset F_1\) and \(A_2 \subset F_2\) such that \(A_1+ F_2= A_2+ F_1\). He also proves that one can take \((A_1,A_2)\) convex (that is, with \(A_1 \cup A_2\) being a convex set) if and only if \((A,B)\) is not minimal.