When are the closed convex subsets countable intersections of halfspaces

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.

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 $\mathbb{R}^n$.