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.
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