Let $X$ be an Alexandrov space with curvature bounded below. For any point $p \in X$, we define $C\_p$ to be the set that consists of all points $q$ such that there are at least two minimizing geodesics from $p$ to $q$.


Can we prove that for any $p \in X$ there exists an open neighborhood $U$ of $p$ such that $U \bigcap C\_p=\emptyset$?



Uniqueness of geodesics in the Alexandrov space
I believe the example of Otsu-Shioya in page 632 of <https://projecteuclid.org/download/pdf_1/euclid.jdg/1214455075> shows $C\_p$ could be dense in the space.


