C OMPOSITIO M ATHEMATICA
V. W. B RYANT
The convexity of the subset space of a metric space
Compositio Mathematica, tome 22, n
o4 (1970), p. 383-385
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383
THE CONVEXITY OF THE SUBSET SPACE OF A METRIC SPACE
by V. W.
Bryant
COMPOSITIO MATHEMATICA, Vol. 22, Fasc. 4, 1970, pag. 383-385 Wolters-Noordhoff Publishing
Printed in the Netherlands
Let
(X, d )
be a metric space and let d * be the Hausdorff metric onthe set X*
consisting
of thenon-empty
closed bounded subsets of X. In this paper we consider what conditions on(X, d )
will ensure that(X*, d*)
ismetrically
convex.DEFINITION.
(i) (X, d)
is(metrically)
convex if for any two distinctpoints x, y
E X there exists zE X,
distinct from themboth,
with(ii) (X, d)
is a(metric)
segment space if for any x, y E X there existsan
isometry f : [0, d(x, y)] ~ (X, d) with f(0)
= xand f(d(x, y))
= y.It is clear that every
segment
space is convex, and it has been shown that the twoconcepts
coincide in acomplete
space(see,
forexample,
[1;
p.41 ]). Now,
if A 5; X and0 ~ ô,
then the setA03B4
is definedby Aa
={x
E X : ~a E A withd(x, a) ~ 03B4}.
We note that in any metric space
A
=~~n=1 A1/n,
and that in anysegment
spaceA03B3+03B4 = (Ay)a
for 0 ~ô.
The Hausdorff metric d * is defined on X*by
It is known that if
(X, d )
iscompact (resp. complete),
then(X*, d*)
iscompact
(resp. complete), proofs
of these resultsbeing
found in[3;
p.38]
and
[2;
p. 29IV].
We are now
ready
toinvestigate
theconvexity
of(X*, d*).
Sinced*({x}, {y})
=d(x, y)
forx, y E X,
it is clear that theconvexity
of(X*, d*) implies
theconvexity
of(X, d).
The theorem below shows when the converseimplication
holds.THEOREM.
If (X, d) is a compact
convex metric space, then so too is(X*, d*).
384
PROOF. In view of the above remarks we need
only
show that(X*, d*)
is convex. Let
A, B ~
X* withd*(A, B) = 03B4
> 0. Thenby
thecompact-
ness of
(X, d) (and
hence ofA, B ) A03B4 = Aa
andB. = B,,.
Since(X, d)
is a
segment
space it follows thatand
similarly B 5; Aa .
Let C =A03B4/2
nB03B4/2.
Then we show thatC =1= cjJ (whence C ~ X*)
and thatd*(A, C ) = d*(C, B ) = 1/2d*(A, B).
Ifa E A 5;
Bc5,
thend(a, b) ~ 03B4
for some b c- B and there exists c e C withd(a, c)
=d(c, b)
=-Ld(a, b) 1/203B4.
Thus c eA03B4/2 ~ B03B4/2 =
C anda
~ C03B4/2.
This shows thatC =1= cjJ
and A 5;C03B4/2,
andsimilarly B ~ C03B4/2.
Thus Ce
X*, d*(A, C) ~ 03B4/2
andd*(C, B) ~ 03B4/2.
But03B4
= d*(A, B) ~ d*(A, C)+d*(C, B) ~ 1/203B4+1/203B4
= ôand so
d*(A, C )
=d*(C, B )
=1/2d*(A, B )
asrequired.
Hence(X*, d*)
is a convex space and the theorem is
proved.
We now
give examples
to show that neither the total-boundedness northe
completeness
of a convex space(X, d ) necessarily implies
the con-vexity
of(X*, d*).
1. Let X be the
subspace
of 3-dimensional EuclideanSpace given by
Then
(X, d )
is atotally
boundedsegment
space.However, by considering
we see that
(X*, d*)
is not convex.2. Let X be the normed vector space of all real null sequences with metric d induced
by
the usual norm. Then(X, d )
is acomplete segment
space and we show that(X*, d*)
is not convex. For ifA =
{{xn} ~ X:xn ~
0 for an odd no.of n, and xn
=1-+1/n whenever Xn :0 0}
E X*B
= {{xn} ~ X:xn ~
0 for an even no.of n, and xn
=1 + 1/n whenever Xn i= 01
EX*,
then
d*(A, B )
= 1 andA1/2 ~ B1/2 = A1 n B1
=~.
Thus there exists noC E X* with
d*(A, C )
=d*(C, B )
=1/2d*(A, B ).
It follows that(X*, d*)
is neither a
segment
space nor a convex space.385
Finally
wegive
anexample
to show that the existence of asegment
from{x}
to{y}
in(X*, d*)
does notimply
the existence of asegment
from x to y in
(X, d).
3. Let X be the subset of
R2 given by X
={(x, y) : Ixl 1, 0 ~ y ~
1and
(x
is rational if andonly if y is)} ~ {(-1, 0), (1, 0)1,
and define ametric d on X
by
d((x, y), (x’, y’)) = 1 y - y’l max (j x 1, |x’|))+|x-x’|,
((x, Y), (x’, y’) EX).
Then
d*({(-1, 0)}, {(1, 0)})
= 2 and themapping f : [0, 2] ~ (X*, d*) given by f(03BB) = {(x, y)
E X : x =03BB-1}
for  e[0, 2]
is anisometry
withf(0) = {(-1, 0)} andf(2)
={(1, 0)}.
This is therefore asegment
between{(-1,0)}
and{(1,0)}
in(X*, d*). However,
there exists nosegment
between
(- 1, 0)
and(1, 0)
in(X, d).
REFERENCES
L. M. BLUMENTHAL
[1] Distance Geometry, Oxford, 1953.
C. KURATOWSKI
[2] Topologie I (2nd edition), Warsaw, 1948.
C. KURATOWSKI
[3] Topologie II, Warsaw, 1950.
(Oblatum 16-111-1970) Dr. V. W. Bryant,
Department of Pure Mathematics,
The University of Shefïield, England.