Reach (mathematics)
Encyclopedia
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.
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn is a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
that roughly describes how curved the boundary
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
of the set is.
Definition
Let X be a subset of Rn. Then reach of X is defined as-
Examples
Shapes that have reach infinity:- single point
- straight line
- full square
- any convex setConvex setIn Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
The graph of ƒ(x) = |x| has reach zero.
A circle of radius r has reach r.