Cut locus

The cut locus is a mathematical structure defined for a closed set ${\displaystyle S}$ in a space ${\displaystyle X}$ as the closure of the set of all points ${\displaystyle p\in X}$ that have two or more distinct shortest paths or geodesics in ${\displaystyle X}$ from ${\displaystyle S}$ to ${\displaystyle p}$. For example, the cut locus of every point on the regular 2-sphere consists of exactly one point, namely the antipodal point.