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Let $X$ be a finite dimensional Alexandrov space with curvature bounded below and non-empty boundary. Let $\gamma$ be a shortest geodesic path in $X$ whose endpoints belong to the interior of $X$.

Is it true that $\gamma$ is contained in the interior of $X$? Is that true at least under the assumption that $X$ is smooth Riemannian manifold with smooth boundary (thus $X$ must be locally geodesically convex)?

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  • $\begingroup$ The case when $X$ is a smooth compact domain in a Riemannian manifold follows from elementary arguments once you know that $X$ is geodesically convex (which follows from the assumption that $X$ is an Alexandrov space as I mention in the comments to Michor's answer). Namely, if $[x,y]$ is a segment in $X$ with $x,y$ in the interior, then the segments from $x$ to points of a small neighborhood of $y$ fill a neighborhood of $(x,y)$, and since by convexity of $X$ all such segments must lie in $X$, no point of $(x,y)$ lies in $\partial X$. $\endgroup$ – Igor Belegradek Jul 14 '20 at 2:29
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No: Take a banana shaped domain with smooth boundary in the plane.

Edit: This answer is wrong. I leave it here for the very instructive comments by Igor Belegradek.

I think I see geodesic triangles near the concave boundary in the banana where Gauss-Bonet implies large negative curvature.

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    $\begingroup$ The banana is not an Alexandrov space. $\endgroup$ – Igor Belegradek Jul 12 '20 at 21:17
  • $\begingroup$ @IgorBelegradek Why is the banana not an Alexandrov space? It seems to me that by compactness, Toponogov's theorem should hold possibly with a very very negative lower curvature bound. $\endgroup$ – Sak Jul 13 '20 at 16:02
  • $\begingroup$ @Sak: see mathoverflow.net/questions/223934/…. $\endgroup$ – Igor Belegradek Jul 13 '20 at 17:34
  • $\begingroup$ @IgorBelegradek wow, the more you know! Thanks! $\endgroup$ – Sak Jul 13 '20 at 18:00
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    $\begingroup$ I wish to add that a smooth compact domain in a Riemannian manifold must be convex if the intrinsic metric on the domain is $CD(K,\infty)$, see Theorem 1.5 in arxiv.org/abs/1902.00942. This applies to every finite-dimensional Alexandrov space by arxiv.org/abs/1003.5948. $\endgroup$ – Igor Belegradek Jul 14 '20 at 1:59

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