This question is edited following the comment of Joseph. He pointed out that the main object of the first version 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 $p \in X$ 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 and piecewise smooth generic boundary, 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 white.
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 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 $M^n\setminus\operatorname{cut-locus}(\partial M^n)$ is canonically foliated by geodesic segments that join $X$ with $\partial M$.
I wonder if this lemma has a name or maybe is contained in some textbook on Riemannian geometry?
Best Answer
Let me continue the comments above here so I can include a figure. Here are examples of the medial axis of two different convex polygons (from my own work):
The term medial axis is used in computer science to denote the same concept as the cut locus.
Franz-Erich Wolter wrote his Ph.D. dissertation on "Cut loci in bordered and unbordered Riemannian manifolds" (Technische Universität Berlin, 1985). That might contain some useful information.