Russian Mathematics (Iz. VUZ)
Vol. 46, No. 1, pp.68{72, 2002
Izvestiya VUZ. Matematika
UDC 515.124.4:514.774
CONTINUITY OF THE METRIC -PROJECTION
ONTO CONVEX SET IN A SPECIAL METRIC SPACE
E.N. Sosov
In this article we prove that certain A.V. Marinov's results on the continuity of metric
-projection onto a convex set in a normed linear space (see 1], 2]) remain valid in a special
metric space.
1. Preliminaries
Let (X ) be a metric space, and S be a set of segments in X (a segment is a curve whose length
is equal to the distance between endpoints of this curve, see 3], p. 42). The elements of S will be
called chords (see 4], p. 23). Assume that X and S satisfy the following conditions.
A. For every chord all its subsegments (the points of chord are included) are chords.
B. Any two points of X can be joined by a unique chord.
C. For all p x y 2 X , we have 2(p !1=2 (x y)) (p x) + (p y), where !1=2 (x y) is the
midpoint of the chord x y] with endpoints x, y.
As is known (see 3], p. 304 4], p. 63), in the direct chord spaces of nonpositive curvature these
conditions hold true. Similar local conditions on a metric space were considered in 5].
We will use the following notation. B x r] (S (x r)) is a closed ball (sphere) of radius r > 0
centered at x xy = (x y), xM = (x M ) for 0, x 2 X , M X , M 6= xM = fy 2 M :
xy xM + g (xM = x0M ) is the metric -projection (the metric projection) of a point x onto a
set M (see 1]) for nonempty sets A B X , (A B ) = supfxB : x 2 Ag is the semi-deviation,
(A B ) = maxf (A B ) (B A)g is the Hausdor distance ( (A B ), (A B ) can be innite) (see
t
1]) (xM 0 F ) = tlim
!+0 (xM F ), where N X (ibid.). Suppose that X satises A and B then
the subset M in X is said to be convex if for any two points x, y in M the chord x y] lies in M .
If X satises A, B, C, then we immediately obtain that
1. For each 2 0 1] and for all p, x, y in X ,
p!(x y) (1 ; )px + py
(1)
where ! (x y) 2 x y] is such that x! (x y) = xy.
2. Every closed (open) ball in X is a convex set.
The following lemma is known.
Lemma 1 (see 1]). Let X be a metric space, x y 2 X , and F , M , N be nonempty subsets
in X , 0. Then (yN F ) (N M ) + (xM1 +0 F ), (F yN ) (M N ) + (F xM2 ;0 ), where
+ xy + yN + (N M ) ; xM , 2 = + yN ; xy ; xM ; (M N ). Here we assume that
1 =
>
0
.
2
Supported by the Russian Foundation for Basic Research, grant no. 00-01-00308.
c
2000
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