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.


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


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.


  • 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, MR 0257325, Zbl 0176.00801