Dini's surface

Template:One source 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

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

Examples

Shapes that have reach infinity:

• 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

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