When are the closed convex subsets countable intersections of halfspaces For what kind of topological vector spaces (separable maybe?) are the closed convex subsets countable intersections of halfspaces. 
I've seen somewhere that it's true for separable Hilbert spaces, but without proof or reference. Is there a reference on this fact (and related questions)? 
 A: This question is addressed in:
Borwein, Jonathan M.; Vanderwerff, Jon D., Constructible convex sets, Set-Valued Anal. 12, No. 1-2, 61-77 (2004). ZBL1059.46056.
A: Those authors have a book on the topic that gives the following more down-to-earth interpretation:
J. M. Borwein and J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples Cambridge University Press, 2010.
https://carma.newcastle.edu.au/resources/jon/Preprints/Books/CUP/cup-final.pdf

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*Just before Proposition 7.5.6 it defines: "A (closed convex) set in a
Banach space is constructible if it is representable as the intersection of countably many closed half spaces."


*Proposition 7.5.6 in the book: "A closed convex subset [of a Banach space] containing the origin is constructible if and only if its polar is weak*-separable."


*The book immediately follows Proposition 7.5.6 with this interpretation: "In particular, all closed convex subsets of a separable [Banach] space are constructible."
The bracketed phrases in the above quotes are my own additions. Since $\mathbb{R}^n$ is a separable Banach space, it works for all closed and convex subsets of $\mathbb{R}^n$.
