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.[1][2]

Formal definition

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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 pM, is the following set-valued function from X to M:

 

Equivalently:

 

The elements in the set   are also called elements of best approximation. This term comes from constrained optimizaiton: we want to find an element nearer to x, under the constraint that the solution must be a subset of M. The function pM is also called an operator of best approximation.[citation needed]

Chebyshev sets

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In general, pM 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 pM 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.[3]

Continuity

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If M is non-empty compact set, then the metric projection pM 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 pM is continuous.[citation needed]

Moreover, if X is a Hilbert space and M is closed and convex, then pM is Lipschitz continuous with Lipschitz constant 1.[citation needed]

Applications

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Metric projections are used both to investigate theoretical questions in functional analysis and for practical approximation methods.[4] They are also used in constrained optimization.[5]

References

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  1. ^ "Metric projection - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-13.
  2. ^ Deutsch, Frank (1982-12-01). "Linear selections for the metric projection". Journal of Functional Analysis. 49 (3): 269–292. doi:10.1016/0022-1236(82)90070-2. ISSN 0022-1236.
  3. ^ "Chebyshev set - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-13.
  4. ^ Alber, Ya I. (1993-11-24), Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications, arXiv:funct-an/9311001, Bibcode:1993funct.an.11001A
  5. ^ Gafni, Eli M.; Bertsekas, Dimitri P. (November 1984). "Two-Metric Projection Methods for Constrained Optimization". SIAM Journal on Control and Optimization. 22 (6): 936–964. doi:10.1137/0322061. ISSN 0363-0129.