Metric projection

In mathematics, a metric projection is a function that maps each element of a metric space to the set of points nearest to that element in some fixed sub-space.{{Cite web |title=Metric projection - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Metric_projection |access-date=2024-06-13 |website=encyclopediaofmath.org}}{{Cite journal |last=Deutsch |first=Frank |date=1982-12-01 |title=Linear selections for the metric projection |url=https://dx.doi.org/10.1016/0022-1236%2882%2990070-2 |journal=Journal of Functional Analysis |volume=49 |issue=3 |pages=269–292 |doi=10.1016/0022-1236(82)90070-2 |issn=0022-1236}}

Formal definition

Formally, let X be a metric space with distance metric d, and let M be a fixed subset of X. Then the metric projection associated with M, denoted p_M, is the following set-valued function from X to M:

$p_M(x) = \arg\min_{y\in M} d(x,y)$

Equivalently:

$p_M(x) = \{y \in M : d(x,y) \leq d(x,y') \forall y'\in M \}
= \{y \in M : d(x,y) = d(x,M) \}$

The elements in the set

\arg\min_{y\in M} d(x,y)

are also called elements of best approximation. This term comes from constrained optimization: we want to find an element nearer to x, under the constraint that the solution must be a subset of M. The function p_M is also called an operator of best approximation.{{citation needed|date= June 2024}}

Chebyshev sets

In general, p_M is set-valued, as for every x, there may be many elements in M that have the same nearest distance to x. In the special case in which p_M is single-valued, the set M is called a Chebyshev set. As an example, if (X,d) is a Euclidean space (Rⁿ with the Euclidean distance), then a set M is a Chebyshev set if and only if it is closed and convex.{{Cite web |title=Chebyshev set - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Chebyshev_set |access-date=2024-06-13 |website=encyclopediaofmath.org}}

Continuity

If M is non-empty compact set, then the metric projection p_M is upper semi-continuous, but might not be lower semi-continuous. But if X is a normed space and M is a finite-dimensional Chebyshev set, then p_M is continuous.{{citation needed|date= June 2024}}

Moreover, if X is a Hilbert space and M is closed and convex, then p_M is Lipschitz continuous with Lipschitz constant 1.{{citation needed|date= June 2024}}

Applications

Metric projections are used both to investigate theoretical questions in functional analysis and for practical approximation methods.{{Citation |last=Alber |first=Ya I. |title=Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications |date=1993-11-24 |arxiv=funct-an/9311001 |bibcode=1993funct.an.11001A }} They are also used in constrained optimization.{{Cite journal |last1=Gafni |first1=Eli M. |last2=Bertsekas |first2=Dimitri P. |date=November 1984 |title=Two-Metric Projection Methods for Constrained Optimization |url=http://epubs.siam.org/doi/10.1137/0322061 |journal=SIAM Journal on Control and Optimization |language=en |volume=22 |issue=6 |pages=936–964 |doi=10.1137/0322061 |issn=0363-0129|hdl=1721.1/2817 |hdl-access=free }}

External links

[https://math.stackexchange.com/q/282034/29780 Do projections onto convex sets always decrease distances?]

References

Category:Approximation theory

Category:Metric spaces