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Let $X$ be an incomplete Alexandrov space with sec $\ge -1$ in the sense that for any point in $X$ there exists a small neighborhood in which the four-points criterion is satisfied.

Suppose $X$ is convex in the sense that any pair of points $p,q \in X$ there exists a minimizing geodesic connecting $p$ and $q$. Can we prove that the completion of $X$ is also an Alexandrov space with sec $\ge -1$.

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  • $\begingroup$ I suppose we can't really use any Gromov-Hausdorff results, because they all assume compactness... $\endgroup$
    – Tom Sharpe
    Commented Apr 22, 2021 at 10:44

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Yes, the conclusion is exactly the main result of Petrunin's paper "A globalization for non-complete but geodesic spaces", Mathematische Annalen volume 366, pages387–393(2016).

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I believe this is true by the work of Nan Li in Theorem A of "Globalization with probabilistic convexity", Journal of Topology and Analysis 07 (2015), no. 04, 719–735.

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