Let $M$ be a Riemannian with nonempty boundary $\partial M$. Define multiplicity of $x\in M$ as the number of minimizing geodesics from $x$ to $\partial M$.
The following fact seems to be standard:
The set of points with multiplicity $\ge 2$ is dense in the cut locus of $M$ with respect to $\partial M$.
Is it proved somewhere?
I know how to prove it, but if you have a one-line proof I am very interested.
Any point of multiplicity 1 in the cut locus is conjugate, but not the other way around. Say for a hemisphere, the cut locus contains a single point — it is a conjugate point of multiplicity $\infty$.