Reach (mathematics) explained

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

reach (X) : = \sup \{r \in R : \forall x \in R ^{n\setminus} X with {\rm dist}(x,X) < r existsauniqueclosestpoint y \in X suchthat {\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.