Let $M$ be a compact, connected $m$-dimensional Riemannian manifold, and let $n\in\mathbb{N}$. Can we always find distinct points $p_1,\dots,p_n\in M$ such that for $i=1,\dots,n$ the regions $A_i=\{p\in M;d(p,p_i)<d(p,p_j)\;\forall j\neq i\}$ all have the same volume?

I'm sure someone has studied this problem before but I haven't found any information about it. I suspect the statement is true, but I have no basis for it.

There are some cases where it is true:

• The case mentioned in the comments: there is an action by isometries on $M$ with some orbit having $n$ points.

• When $n=2$ the statement seems to be a consequence of the intermediate value theorem (consider two disjoint arcs joining $p_1$ and $p_2$ and swap them using the paths). In general assuming that the measures $\mu(A_1),\dots,\mu(A_n)$ depend continuously on $p_1,\dots,p_n$ maybe some argument using homology (degree theory/Brouwer's fixed point theorem/something like that) could work, but I have not found anything.

In case we can always find the $n$ points, here is a stronger version of the question:

Given some nonempty open set $U\subseteq M$, can we choose points $p_1,\dots,p_n\in U$ that divide $M$ into regions of equal volume?