View source for Metric projection

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''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|Element of best approximation]]) from the set  $  M \subset  X $. 
If  $  M $
is a [[Chebyshev set|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Singer,   "The theory of best approximation and functional analysis" , ''CBMS Regional Conf. Ser.'' , '''13''' , SIAM  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Berdyshev,   "Uniform continuity of the metric projection and of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m06367020.png" />-projection" , ''Theory of Approximation of Functions. Internat. Conf., Kaluga 1975'' , Moscow  (1977)  pp. 37–41  (In Russian)</TD></TR></table>