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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)?

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2 Answers 2

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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.

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  • $\begingroup$ I pulled up this paper but I could not discern a clear answer. Prop 3.3 seems the closest match and says that if some $B_{X^*}$ thingy satisfies a Corson compact thingy then "every closed convex subset in X is constructible if and only if there exists at least one closed bounded convex constructible set in X." I'm not sure what other (Banach space?) assumptions X is assumed to have. I don't even know if $\mathbb{R}^n$ satisfies the Corson-and-$B_{X^*}$ thingy. This paper could have been more helpful if the authors had written a few sentences to give down-to-earth interpretations. $\endgroup$
    – Michael
    Jul 9, 2020 at 22:55
  • $\begingroup$ Note to functional analysis mathematicians: Please tell us if your results hold for $\mathbb{R}^n$. Anyway I can likely prove the $\mathbb{R}^n$ case myself but it would have been nice to have a reference, at least the case when a set is closed, bounded, and convex in $\mathbb{R}^n$ it seems easy-to-prove (the case of just closed and convex seems more tricky). $\endgroup$
    – Michael
    Jul 9, 2020 at 22:58
  • $\begingroup$ On the positive side, the paper seems to be well written with a simple and easy flow, it just uses a lot of loaded terminology that is hard for me to plug into. I would have to read the paper in detail from start to finish, looking up on wikipedia every 5 minutes to get the different terminology, to be confident in applying the results even to basic spaces like $\mathbb{R}^n$. $\endgroup$
    – Michael
    Jul 9, 2020 at 23:08
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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

  • 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 $\mathbb{R}^n$.

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