Let $(X,d)$ be an $n$-dimensional $(n< \infty)$ complete geodesic metric space, where any two points in $X$ are joined by a unique shortest geodesic. Let $S$ be a sufficiently small metric $(n-1)$-sphere in $X$. Let $\epsilon>0$ be the radius of $S$. Pick a point $p \in S$. Find a metric $(n-1)$-sphere $S'$ of radius $0<r < \epsilon/2$ around $p$. A $k$-dimensional metric sphere $S^k$ in $(X,d)$ of radius $r$ centered at $p$ is given by a $k$-dimensional subset of $(X,d)$: $S^k \subset \{x\in X|d(p,x) = r\}$. Here dimension I'm referring to covering dimension.


Is the intersection $S \cap S'$ always an $(n-2)$-dimensional metric sphere?



Is the intersection of two distinct sufficiently small metric spheres always empty, a point or a metric sphere of lower dimension?
Let $C\_\alpha$ denote the $n$-dimensional closed Euclidean solid cone with the cone angle $\alpha\in (0, \pi)$. Let $X\_\alpha$ be the metric space obtained by gluing two copies $C^\pm\_\alpha$ of $C\_\alpha$ at their tips, and equipped with the natural path-metric. Let $o\in X\_\alpha$ denote the common tip of the cones. I will use it as the center of the first sphere $S=S(o,1)$; I will use $\epsilon=1$. For every point $x\in S$, the sphere $S'=S(x,r)$ is $n-1$-dimensional. However, for each $r\in (0,1)$, for all sufficiently small $\alpha$, for every $x\in S\subset X\_\alpha$, the intersection $S'\cap S$ is empty, hence, has dimension $-1$, not $n-2$. By modifying this construction one can build uniquely geodesic spaces where spheres as in your question are $n-1$-dimensional but have arbitrary dimension of their intersection, between $-1$ and $n-1$. 


**Edit.** Here is a generalization, to ensure that the spheres $S, S'$ are connected (and locally path-connected) and the intersection $S\cap S'$ is nonempty. 


Let $Y$ be a closed solid cone in the Euclidean space $E^k, 1\le k<n$, with the tip $o$. I will glue $Y$ to the space $X\_\alpha$ as above so that $o$ is identified with the common tip $o\in X$ and two boundary rays of $Y$ are identified with geodesic rays in the two cones $C^\pm\_\alpha \subset X$. Let $Z$ denote the resulting path-metric space. It is easy to check (say, using [Reshetnyak gluing theorem](https://en.wikipedia.org/wiki/Reshetnyak_gluing_theorem)) that $Z$ is a $CAT(0)$ space, hence, is uniquely geodesic. 


Moreover, both spheres $S, S'\subset Z$ (defined as before) are connected and locally path-connected. Furthermore, 
for every $r$, there is $\alpha$ such that $S\cap S'$ is $(k-2)$-dimensional. (Note that $k-2<n-2$.) 


By working a bit harder, one can modify this construction so that spheres are still connected and locally path-connected, while $S\cap S'$ is $(n-1)$-dimensional, where $n$ is the dimension of the ambient space. 


