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?


**Comments**


* 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 $\infty$.



Conjugate points on cut locus
*I've got a letter from Stephanie Alexander with a complete answer. Let me summarize it here.*


The first proof is given in 4.8 of "Schnittort und konvexe Mengen..." by Hermann Karcher (1968). (The formulation is slightly weaker, but from the proof proves our statement follows; the idea is the same as in the answer of Ian Agol.)


An other proof was given in "Decomposition of cut loci" by Richard Bishop (1977).


Latter it appears as Lemma 2 in "Distance function and cut loci..." by Franz-Erich Wolter (1979). The proof is basically
the same as Karcher's.


