Let $X$ be a Banach space $\{C_n\colon n\in\mathbb N\}$ a collection ofconvex sets in $X$. Is the set $$C=\bigcap_{n\in\mathbb N}C_n$$ convex?
Are countable intersections of convex sets convex?
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functional-analysis
banach-spaces
convex-analysis
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2Taking intersections is very well behaved. Most properties are preserved. Just appeal to the definition. If $x,y\in C$ then $x,y\in C_n$ for each $n\in\mathbb N$ and use the convexity of the $C_n$s – 2012-07-14
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2Yes: pick two points in the intersection $x$ and $y$. For each $n$ and $0\leq a\leq 1, $ax+(1-a)y\in C_n$. (actually it works for arbitrary intersections. – 2012-07-14
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0They don't even have to be countable. I'm tempted to say even a proper class, as opposed to a set, of convex sets would have a convex intersection. But I don't know where you'd find any such class that's not a set. – 2012-07-14