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Let $\eta$ be a closed convex subset of ${\mathbb R}^n$ of convex dimension $n$ (not necessarily compact) with nonempty boundary $\partial \eta$. Then $\eta$ is an $n$-dimensional topological manifold with boundary and therefore $\partial \eta$ of $\eta$ is an $(n-1)$-dimensional submanifold of ${\mathbb R}^n$. Suppose that $\partial \eta$ is contained in an algebraic hypersurface in ${\mathbb R}^n$, so that the Zariski closure $V$ of $\partial \eta$ is not all of $\mathbb R^n$. Does it follow that all of the irreducible components of $V$ are $(n-1)$-dimensional as irreducible subvarieties of $\mathbb R^n$? (Equivalently, must the ideal of all polynomials in $\mathbb R[X_1, X_2, \ldots, X_n]$ vanishing on $\partial \eta$ be principal?) Also, must $\partial \eta$ be a CW-complex?

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The answer to the first question is indeed yes by standard results in semi-algebraic geometry. A reference is Lemma 2.4 and Proposition 2.7 in http://arxiv.org/pdf/1405.7822.pdf

The answer to the second question might also be yes. Does it not follow from the fact that every semi-algebraic set can be triangulated? A reference for this statement is the textbook "Real Algebraic Geometry" by Bochnak, Coste, and Roy (chapter 7).

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