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.

References

[1]	I.M. Singer, "The theory of best approximation and functional analysis" , CBMS Regional Conf. Ser. , 13 , SIAM (1974)
[2]	L.P. Vlasov, "Approximative properties of sets in normed linear spaces" Russian Math. Surveys , 28 : 6 (1973) pp. 3–66 Uspekhi Mat. Nauk , 28 : 6 (1973) pp. 3–66
[3]	V.I. Berdyshev, "Uniform continuity of the metric projection and of the -projection" , Theory of Approximation of Functions. Internat. Conf., Kaluga 1975 , Moscow (1977) pp. 37–41 (In Russian)