On the closedness of the algebraic difference of closed convex sets

We characterize in a reflexive Banach space all the closed convex sets C₁ containing no lines for which the condition C₁^∞∩C₂^∞={0} ensures the closedness of the algebraic difference C₁−C₂ for all closed convex sets C₂. We also answer a closely related problem: determine all the pairs C₁, C₂ of closed convex sets containing no lines such that the algebraic difference of any sufficiently small uniform perturbations of C₁ and C₂ remains closed. As an application, we state the broadest setting for the strict separation theorem in a reflexive Banach space.