Suppose $X$ is a CAT(0) space and $\gamma$ be a geodesic joining two point say $x$ and $y$ in $X$.Then, can we find a small ball around $y$ such that geodesic joining the point $x$ and a point of ball are concatenation of geodesic $\gamma$ and a geodesic contained in that ball?
i am thinking the following: Since balls in CAT(0) spaces are convex and local geodesics are geodesics so if take a point in ball then only problem will be at point $y$ whether it is local geodesic at $y$ or not but if it is not local geodesic then there will be some geodesic that will still be inside the ball as balls are convex but how shall we choose radius of the ball? Can we do the same for large radius ball also?
Here's a counterexample: $\mathbb R^2$.