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If A and B are nonempty convex sets. And C = A + B. How to prove int(C) = int(A) + int(B)?
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Have you tried some concerete examples?kaosAD said:This is what I have tried a some what modified problem. But it goes nowhere.
I will use 'cl' to mean closure and 'bd' to mean boundary. Assume A = cl(A) and B = cl(B) to make life easier for me. Let [tex]\bar{x} \in \textup{bd}(A)[/tex]. Consider a sequence [tex]\{x_k\}[/tex] belonging to int(A) and converging to a limit point [tex]\bar{x}[/tex]. Pick any [tex]y \in B[/tex]. Hence the sequence [tex]\{x_k + y\}[/tex] converges to some point, say [tex]\bar{z}[/tex]. This [tex]\bar{z}[/tex] may or may not belong to int(C). I reckon if [tex]y \in \textup{bd}(B)[/tex], then the [tex]\bar{z}[/tex] does not belong to int(C).
I don't think it goes well following this line of argument. Need some help.