Let $D \subset \mathbb{R}^n$ be a bounded domain. If $\partial D$ is a real-analytic sub-manifold, it is not very difficult to show that $\partial D$ does not contain any line segments. Now, assume that $\partial D$ is a finite disjoint union of (connected) real-analytic sub-manifolds of $\mathbb{R}^n$ such that there is precisely one sub-manifold of dimension $n-1$, say $B$. Is it true that $B$ does not contain any line segments?
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