On pairs of closed bounded convex sets that lie in complementary subspaces

Let X be a topological vector space over the field R. Let B(X) be the family of all nonempty closed bounded convex subsets of X, /C(X) = {A e B(X) | A is compact}.

Let A + B = {a + b | a e A, b e B] be the Minkowski sum of A and B in B(X). It is well known that /C(X) with Minkowski sum is the semigroup that satisfies the law

4c s|c of cancellation [7]. If we define A+B = A -f B then also B(X) with addition 4-is the semigroup satisfying the law of cancellation [7].