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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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Are you still assuming each of the components of the disjoint union are real analytic? Are you assuming anything else about $D$? As stated, the question looks trivially false to me. – Willie Wong Mar 12 '12 at 10:25
Elaborating on "trivially false", consider the domain $(x-(1-z^2))^2+y^2\le (1-z^2)^2$, $-1\le z\le 1$. – fedja Mar 12 '12 at 19:52
@willie Wong Yes I am assuming that each component is real-analytic. – Jaikrishnan Mar 13 '12 at 3:12

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