I would like to use some "intuitively clear" properties of Voronoi cells in general Riemannian manifolds, but I have trouble finding references.

Let $(X,d)$ be a connected Riemannian manifold and let $S$ be a discrete set of points. Define $Vor(s,S) = \{x\in X\colon \text{for any $s'\in S$ we have } d(x,s) \le d(x,s')\}$.

Assume $S$ is such that for any $s\in S$ there is a finite subset $N(s)\subset S$ such that $Vor(s,S) = Vor(s,N(s))$.

Question 1. Is it true that $Vor(s,S)$ are submanifolds with a boundary, and the intersection of any number of them is a submanifold with a boundary?

Question 2. Under what general conditions $Vor(s,S)$ induce a structure of a CW-complex on $X$?

In general, define the dual $2$-complex $\hat X_2$: the vertices are $S$, there is an edge between $s$ and $s'$ if $Vor(s,S)\cap Vor(s',S)$ is of codimension $1$ in X and there is a face spanned by a finite subset $F \subset S$ if $F$ is a maximal subset such that the cells $Vor(f,S), f\in F$ share a codimension-2 submanifold.

Perhaps this ad-hoc definition of the dual complex is obviously flawed, in which case I'd appreciate references to a better one. To me it is not even immediately clear $\hat X_2$ is a CW complex i.e. if the boundary of a face is in the $1$-skeleton.

Question 3. Is it the case that if $X$ is contractible then $\pi_1(\hat X_2) =\{1\}$? (or if not, under what assumptions is it the case?)

Remark: In the previous version of Question 1 I asked whether the intersections are a connected manifolds. Vidit Nanda below provided an example which shows that it's not always the case

Handbook of Combinatorics, Vol.2, Ch. 34, Theorem 10.7 (tinyurl.com/o8n7xll) $\endgroup$1more comment