Reach (mathematics)

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In mathematics, the reach of a subset of Euclidean space Rⁿ is a real number that roughly describes how curved the boundary of the set is.

Definition

Let X be a subset of Rⁿ. Then 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:

single point
straight line
full square
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, series Die Grundlehren der mathematischen Wissenschaften, Band 153, New York: Springer-Verlag New York Inc., pp. xiv+676, ISBN 978-3-540-60656-7, MR 0257325

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