Reach (mathematics)

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


 * $$\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)\}. $$

Examples
Shapes that have reach infinity include
 * a single point,
 * a straight line,
 * a full square, and
 * any convex set.

The graph of ƒ(x) = |x| has reach zero.

A circle of radius r has reach r.