This question is edited following the comment of Joseph. He pointed out that the main object of the first vesrion of this question is the cut locus.
Recall that the cut locus of a set $S$ in a geodesic space $X$ is the closure of the set of all points $p$ that have two or more distinct shortest paths in $X$ from $S$ to $p$. http://en.wikipedia.org/wiki/Cut_locus
A simple lemma shows that for a disk $D^2$ with a Riemannian metric the cut locus of $D^2$ with respect to its boundary is a tree. A picture of such tree can be found on page 542, figure 17 of the article of Thurston "Shapes of polyhedra". The tree is of white colour. http://arxiv.org/PS_cache/math/pdf/9801/9801088v2.pdf For an ellipse on the 2-plane the tree is the segment that joins its focal points.
More generically for a Riemannian manifold $M^n$ with a boundary the cut locus of $\partial M$ should be a deformation retract of $M$ (I guess it is a $CW$ complex of dimension less than $n$). To prove this lemma notice that $M^n\setminus Cat locus(\partial M^n)$ is canonically foliated by geodesic segments that join $X$ with $\partial M$.
I wonder if this lemma has a name or maybe it is contained in some textbook on Riemannian geometry?