Is any geodesic in the tangent cone of an Alexandrov space a limit geodesic?

Question

If $X$ is an Alexandrov space of curvature bounded below by a real number $k$, is it true that any geodesic in the tangent cone $T_pX$ can be realized as a limit of geodesics when we view $T_pX$ as the Gromov-Hausdorff limit of rescalings of neighborhoods of $p$?

More generally, if $X_i$ is a sequence of Alexandrov spaces with curvatures bounded below uniformly by $k$ and $\dim X_i=n$, such that $X_i$ converges in the Gromov-Hausdorff sense to an Alexandrov space $X$ with $\dim X=n$ (i.e., the sequence is non-collapsing), is it true that any geodesic in $X$ is the limit of a sequence of geodesics in $X_i$?

Answer 1

The answer is "yes" assuming you are interested in minimizing geodesics. Moreover the statement holds in the collapsing case as well.

Fix two points $p$ and $q$ in the limit space $X$. If there is unique geodesic $[p,q]$ connecting these two points then we can choose arbitrary converging sequences $X_n\ni p_n\to p$ and $X_n\ni q_n\to q$ --- any sequence of geodesics $\gamma_n=[p_n,q_n]$ does the trick.

If there are many geodesics, fix $p',q'\in [p,q]$ close to the ends. By comparison, the geodesic $[p',q']$ is uniquely defined. Therefore by the argument above $[p',q']$ is a limit of geodesics in $X_n$.

Repeat this argument for a sequence of pairs of points $p'_k\to p$ and $q'_k\to q$. You get double sequence of geodesics $\gamma_{n,k}$ in $X_n$. Applying diagonal procedure you get an approximation of $[p,q]$ by a sequence $\gamma_{n,k(n)}$.