Approximations and expansions Definition

Chebyshev set

Builds on Normed vector space Best approximation

Definition

Let $X$ be a normed vector space and let $K \subseteq X$ be nonempty. The set $K$ is a Chebyshev set if every $x \in X$ has exactly one best approximation from $K$ , that is, exactly one $k \in K$ with $\norm{x - k} = \inf_{y \in K} \norm{x - y}$ . The map $P_K \colon X \to K$ sending $x$ to this point is the metric projection onto $K$ .

Remarks

Example: in $\R^2$ with the Euclidean norm, the closed unit disc $D$ is a Chebyshev set. For $\norm{x} \le 1$ the nearest point is $x$ itself. For $\norm{x} > 1$ it is $x / \norm{x}$ and no other point of $D$ , since the triangle inequality is strict off the ray through $x$ .

Non-example: the unit circle in $\R^2$ is not a Chebyshev set, because the origin has every point of the circle as a nearest point.

A Chebyshev set is closed, since a point of the closure not in $K$ would have distance $0$ but no nearest point. Every nonempty closed convex set in a Hilbert space is a Chebyshev set, by the Hilbert projection theorem. In $\R^n$ with the Euclidean norm the converse holds: every Chebyshev set is convex. Whether every Chebyshev set in an infinite-dimensional Hilbert space is convex is an open problem.

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Tag	`N9QDDYY7GK`
URI	`m2://N9QDDYY7GK`
Kind	definition
Writing	W0 stub
MSC	41A65, 41A52, 46B20
Defines	Chebyshev set
Uses	Normed vector space, Best approximation

Chebyshev set

Definition

Remarks

Reading path

Further reading

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