Let $p,q\in M$, $p\neq q$, where $M$ is a Riemannian Manifold. We will let the bisector of $p,q$ be $\mathcal{B}(p,q)=\{x\in M;d(p,x)=d(q,x)\}$. Does $\mathcal{B}(p,q)$ have measure $0$?

I was thinking about this problem with a colleague and we believe we can prove it when $M$ is a closed manifold, but even in that case the proof was much more complicated than expected. I encountered this problem while thinking about this question.