Reach (mathematics)

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

reach(X):=sup⁡{r∈ℝ:∀x∈ℝn∖X with dist(x,X)<r exists a unique closest point y∈X such that dist(x,y)=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.

## References

* Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 153, New York: Springer-Verlag New York Inc., pp. xiv+676, ISBN 978-3-540-60656-7

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