Supriya Ghosh (Editor)

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.

Contents

Definition

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

reach ( X ) := sup { r ∈ R : ∀ x ∈ R n ∖ X  with  d i s t ( x , X ) < r  exists a unique closest point  y ∈ X  such that  d i s t ( x , y ) = d i s 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

    Reach (mathematics) Wikipedia
    (Text) CC BY-SA