Let ''X'' be a subset of R^''n''. Then the reach of ''X'' is defined as :

\text(X) := 
    \sup \.

Examples

Shapes that have reach infinity include * a single point, * a straight line, * a full square, and * any

convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...

. The graph of ''ƒ''(''x'') = , ''x'', has reach zero. A circle of radius ''r'' has reach ''r''.

References

* {{citation , last = Federer , first = Herbert , authorlink = Herbert Federer , title = Geometric measure theory , publisher = Springer-Verlag New York Inc. , location = New York , year = 1969 , pages = xiv+676 , isbn = 978-3-540-60656-7 , zbl=0176.00801 , mr=0257325 , series = Die Grundlehren der mathematischen Wissenschaften , volume = 153 Geometric measurement Real analysis Topology