Cut locus

The cut locus is a mathematical structure defined for a closed set formula_1 in a space formula_2as the closure of the set of all points formula_3 that have two or more distinct shortest paths in formula_2 from formula_1 to formula_6. 
Definition in a special case.
Let formula_2 be a metric space, equipped with the metric formula_8, and let formula_9 be a point. The cut locus of formula_10 in formula_2 (formula_12), is the locus of all the points in formula_2 for which there exists at least two distinct shortest paths to formula_10 in formula_2. More formally, formula_16 for a point formula_17 in formula_2 if and only if there exists two paths formula_19 such that formula_20, formula_21, formula_22, and the trajectories of the two paths are distinct.
Examples.
For example, let "S" be the boundary of a simple polygon, and "X" the interior of the polygon.
Then the cut locus is the medial axis of the polygon. The points on the medial axis are centers of maximal disks that touch the polygon boundary at two or more points, corresponding to two or more
shortest paths to the disk center.
As a second example, let "S" be a point "x" on the surface of a convex polyhedron "P", and "X" the surface itself. Then the cut locus of "x" is what is known as the ridge tree of "P" with respect to "x". This ridge tree has the property that cutting the surface along its edges unfolds "P" to a simple planar polygon. This polygon can be viewed as a net for the polyhedron.
Example for the special case.
Let formula_23, that is the regular 2-sphere. Then the cut locus of every point on the sphere consists of exactly one point, namely the antipodal one.