Reach (mathematics)

Let X be a subset of Rⁿ. Then the reach of X is defined as

${\displaystyle {\text{reach}}(X):=\sup\{r\in \mathbb {R}$ :\forall x\in \mathbb {R} ^{n}\setminus X{\text{ with }}{\rm {dist}}(x,X)<r{\text{ exists a unique closest point }}y\in X{\text{ such that }}{\rm {dist}}(x,y)={\rm {dist}}(x,X)\}.}