User:Zfeinst/Closure of Sum of Closed Sets

If $$A,B \subseteq X$$ are closed sets in a topological vector space then $$A + B$$ is closed if
 * 1) Either $$A$$ or $$B$$ is compact set
 * 2) Dieudonne's theorem: Let nonempty closed convex sets $$A,B \subset X$$ a locally convex space, if either $$A$$ or $$B$$ is locally compact and $$\operatorname{recc}(A) \cap \operatorname{recc}(B)$$ (where $$\operatorname{recc}$$ gives the recession cone) is a linear subspace, then $$A - B$$ is closed.
 * 3) Let nonempty closed convex sets $$A,B \subset \mathbb{R}^d$$ such that for any $$y \in \operatorname{recc}(A) \backslash \{0\}$$ then $$-y \not\in \operatorname{recc}(B)$$, then $$A + B$$ is closed.
 * 4) Let nonempty closed convex sets $$A,B \subset X$$ a reflexive Banach space contain no lines, if $$\lim_{r \to \infty} \inf \{\|a - b\|: a \in A \backslash \mathcal{B}(r), b \in B \backslash \mathcal{B}(r)\} = \infty$$ then $$A - B$$ is closed (where $$\mathcal{B}(r) = \{x \in X: \|x\| \leq r\}$$).  In fact, if this condition is satisfied then any two closed convex uniform perturbations $$A_{\epsilon}$$ and $$B_{\epsilon}$$ fulfilling $$A_{\epsilon} \subseteq A + \mathcal{B}(\epsilon), \; A \subseteq A_{\epsilon} + \mathcal{B}(\epsilon)$$ and $$B_{\epsilon} \subseteq B + \mathcal{B}(\epsilon), \; B \subseteq B_{\epsilon} + \mathcal{B}(\epsilon)$$ then $$A_{\epsilon} - B_{\epsilon}$$ is closed.