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 $\ge -1$.



Completion of an Alexandrov space
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).


