# Metric projection

operator of best approximation

A many-valued mapping $P _ {M} : x \rightarrow P _ {M} x$, associating to each element $x$ of a metric space $X = ( X , \rho )$ the set

$$P _ {M} x = \ \{ {m \in M } : {\rho ( x , m ) = \rho ( x , M ) } \}$$

of elements of best approximation (cf. Element of best approximation) from the set $M \subset X$. If $M$ is a Chebyshev set, then the metric projection is a single-valued mapping. The problem of constructing an element of best approximation is often solved approximately, that is, an element is determined in the set

$$P _ {M} ^ {t} x = \ \{ {m \in M } : {\rho ( x , m ) \leq t + \rho ( x , M ) } \} ,$$

where $t > 0$ is sufficiently small. From the properties of the mapping $P _ {M} ^ {t} : x \rightarrow P _ {M} ^ {t} x$ it is sometimes possible to obtain properties of the set $M$. E.g., if for any element $x$ of a normed space $X$ a number $t = t ( x) > 0$ exists such that $P _ {M} ^ {t} x$ is convex (connected), then $M$ is convex (respectively, connected).

From the point of view of applications it is useful to know whether the metric projection has such properties as linearity, continuity, uniform continuity, etc. A metric projection on a Chebyshev subspace of a normed space is, in general, not linear. If the metric projection on each subspace of fixed dimension is single-valued and linear, then $X$ is linearly isometric to an inner-product space. The metric projection on a non-empty approximately-compact set in a metric space is upper semi-continuous; in particular, in a normed space the metric projection onto a finite-dimensional Chebyshev subspace is continuous; the metric projection may be not lower semi-continuous if the subspace is not Chebyshev. There exists a reflexive strictly-convex space and an infinite-dimensional subspace on which the metric projection is discontinuous. The metric projection on any closed convex set $M$ in a Hilbert space satisfies a Lipschitz condition:

$$\| P _ {M} x - P _ {M} y \| \leq K \| x - y \| ,$$

with constant $K = 1$.

The continuity property of a metric projection and its generalizations have found applications in ill-posed problems, in the convexity problem for Chebyshev sets, in the construction of elements of best approximation, etc.

How to Cite This Entry:
Metric projection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Metric_projection&oldid=47830
This article was adapted from an original article by V.I. Berdyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article