For a Riemannian manifold $M$ the cut locus of any point $p$, denoted by $\mathrm{Cu}(p)$, consists of all points $q\in M$ such that if there exists a distance minimal geodesic joining $p$ to $q$, then any extension of it beyond $q$ is not minimal. For example, the cut locus of any point in a sphere will be the corresponding antipodal point.
The separating set of a point $p$, denoted by $\mathrm{Se}(p)$, is the collection of all points $q$ such that there exists at least two minimal geodesic joining $p$ to $q$.
By distance minimal, I mean the length of the geodesic is same as the distance between the points.
A theorem(see Lemma 2) of Wolter says that the closure of the set $\mathrm{Se}(p)$ is the cut locus $\mathrm{Cu}(p)$.
My question: What are some examples where $\mathrm{Se}(p) \subsetneq \mathrm{Cu}(p)$. In most of the classical examples both of the sets are same.
