Reach (mathematics)

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

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

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