In mathematics, a metric projection is a function that maps each element of a

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

to the set of points nearest to that element in some fixed sub-space.

Formal definition

Formally, let ''X'' be a

with

distance Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...

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'':

$p_M(x) = \arg\min_ d(x,y)$

Equivalently:

$p_M(x) = \
= \$

The elements in the set

\arg\min_ d(x,y)

are also called elements of best approximation. This term comes from

constrained optimization In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. The obj ...

: 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.

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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

(Rⁿ with the

Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...

), then a set ''M'' is a Chebyshev set if and only if it is closed and

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

Continuity

If ''M'' is non-empty

compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...

, 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. Moreover, if X is a

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

and M is closed and convex, then ''p_M'' is

Lipschitz continuous In mathematical analysis, Lipschitz continuity, named after Germany, German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in h ...

with Lipschitz constant 1.

Applications

Metric projections are used both to investigate theoretical questions in

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...

and for practical approximation methods. They are also used in

External links

Do projections onto convex sets always decrease distances?

References

{{Metric spaces Approximation theory Metric spaces