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. In case it is true, we can ask a stronger version:

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