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Let $\{M_i\}_{i=1}^\infty$ be a sequence of (compact) Moebius bands with Riemannian metrics with Gauss curvature at least $-1$ and such that the boundaries are geodesically convex (equivalently, the second fundamental forms of $\partial M_i$ are non-negative).

Question. Can $\{M_i\}$ converge in the Gromov-Hausdorff metric to a segment?

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    $\begingroup$ Suppose your Moebius band is a rectangle $[0,1]\times[0,a]$ with glued $[0,1]\times\{0\}$ and $[0,1]\times\{a\}$. By sending $a\to 0$, you get a collapse to $[0,\tfrac12]$. $\endgroup$
    – Anton Petrunin
    Commented Sep 30 at 19:14
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    $\begingroup$ @Petrunin: Oops, you are right. Thank you. This is the final answer. $\endgroup$
    – asv
    Commented Sep 30 at 19:31

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