MACHADO, H.

V
KODAI MATH. SEM. REP
25 (1973). 307-320

A CHARACTERIZATION OF CONVEX SUBSETS OF

NORMED SPACES

BY HILTON VIEIRA MACHADO

1. Introduction.

There are several ways in which one can introduce a notion of convexity in
a metric space (see e.g. [1], [2], [4] and [5]). This paper deals with that of W-
convexity as introduced by Takahashi [8].
PF-spaces are natural set-ups for certain generalizations of fixed-point theorems
for nonexpansive mappings in Banach spaces ([3], [6] and others), some of these
generalizations appear in [7]. Thus it seems desirable to stablish the relationship
between ^-convexity and the usual notion of (linear) convexity. Theorems 1 and
2, below, give an answer to this problem in terms of simple geometrical condi-
tions. Roughly speaking, Theorem 1 tells us when a PF-space X is, essentially, a
convex subset of some normed space E. Theorem 2 says that the space E is
essentially unique.

2. Basic properties.

In [8] Takahashi introduced the notion of convexity in a metric space X by

means of an operator W from X2 x [0, 1] into X, such that

(A) d[z, W(x, y, a)] ^ ad[z, x] + (1 - a)d[z, y]

for all x, y and z in X and every a in the interval /=[0,1]. Here we refer to
such a space as a PF-space and, for simplicity, write (x, y, a) instead of W(x, y, a).
It is clear that the usual notion of (linear) convexity in a normed space is of this
type with (x,y,a)=ax+(l—a)y. As in the linear case, we say that a subset Y of
a TF-space X is W-convex if (x, y, a) belongs to Y whenever x and y are elements
of F, and a is in the unit interval.
A nontrivial example of a IF-space is obtained as follows: consider a closed
subset X of the unit ball Si = {||#|| = l} in a Hubert space H, such that X has
diameter 3(X)^Λ/~2 and is geodesically connected, i.e., the point

W(x,y,a) =

Received September 13, 1972.

307
308 HILTON VIEIRA MACHADO

lies in X whenever x,y$X znά <χ€l. The metric space we obtain by measuring
distances in X through central angles, i.e., with the metric

di[x, y]=co$~1(x, y), V x, yζX,

turns out to be a ΐf-space (whose convex sets are exactly the geodesically con-
nected subsets of X).
The following properties are direct consequences of (A):

(1) d[x,(x,y,a)] = (l-a)d[x,y].

( 2 ) d[y, (x, y, a)]=ad[x, y].
( 3 ) (x, x, a) = (x, y, l) = (y, x, 0) = x.
(4 ) α€[0,1] -> {x, y,a)sX is an injective mapping.
(5) balls (open or closed) in X are TF-convex sets.
(6) the union of a directed family of Tf-convex sets and the intersection of
TF-convex sets are Tf-convex.
Motivated by the linear case we define multiple convex combinations, induc-
tively: if xu -"yXn^X and au •• ,α w e/, with ΣΓ«i = l» we set
a i
( - \—(( <*Π-1 \ -, \ -r -,

and

(Xlf ~',Xn\ 0, —,l) = Xn.

We now introduce two other conditions (both are satisfied in normed spaces):

(B) (x,y,z; a,β9γ) = (y,z,x; β,γ,a), Vx,y,zGX, a,β,γ£l, a + β + γ = l.

(C) d[(x, z, α), (y, z} a)]=ad[x, y], Vxyy,zεX and a€Ϊ.

Their geometrical meaning is clear: the first says that convex combinations do
not depend of the order they are carried out, the second says that the distance is
homothetic.
At this point we can state our main result:

THEOREM 1. If X is a W-convex metric space satisfying conditions (B) and (C)

above, then there is an isometry 3 from X onto a convex subset of some normed
space Ey which preserves convex combinations, i.e., for every x,y£X and a£l we
have that

If X is a metric space with an operator W: X2 x [0,1] -> X, it is easy to see

 CONVEX SUBSETS OF NORMED SPACES 309

that (A), (B) and (C) are necessary conditions for the existence of an isometric
embedding, preserving convex combinations, of X in some normed space E.
Theorem 1 tells us that the three conditions are also sufficient, i.e. they charac-
terize those TF"-spaces which are, essentially, convex subsets of normed spaces,
since then the metric and convexity structures coincide. In this way Theorem 1
gives a characterization of convex subsets in normed spaces.
The construction of the supporting space E is somewhat technical but simple
to outline: heuristically the first step involves the construction of the cone C=R+X
of all "positive multiples" of elements of X, the normed space E is then defined
as the set C—C of all "differences" of elements in C.
To prove Theorem 1 we will need some auxiliary identities — if X is a W-
convex space satisfying conditions (B) and (C) we have:

(7) (fa y, a), z, β)=((y, z, ^ " ^ ), x, l-aβj if aβ*l.

(8) (x,y,a) = (y,xfl-a).
(9) ((x,y,a),y,β) = (x,y,aβ).
(10) if (xtz,a) = (y,z,a) and α^O then x=y.
(11) d[(x9y,a), (x,y,β)] = \a-β\d[x,yl
(12) if σ is a permutation of the set {1,2, •••,»} then

Proofs. For (7) just consider the identity

(a?, y, z\ aβ, β(l-a), l-β) = (y, z, x\ j9(l-α), 1-jS, aβ).

If β^ψθ} we get, by the definition of multiple convex combinations, formula (7).

If /3=0, (7) follows immediately from (3).

(8) Assuming α ^ l w e have, by (7), that

(x, y, «) = ((#, y, α), x, l) = ((y, x, 1), x, l—a) = (y, x, I-a)

If α = l, (8) follows from (3).

(9) is a direct consequence of (7) and (8).

(10) follows from condition (C).
(11) if β^a*0 we have by (10) that

d [ ( a ? , yy β), (x, y , a ) ] = d \ ( ( x , y , ^ ) , y 9 a j , (x, y , a ) \

, y,
310 HILTON VIEIRA MACHADO

The other case, a=β=0 is trivial.

(12) it is enough to consider the case when σ is a transposition, the proof by
induction is straightforward and because of that is ommited.

3. The cone extension.

In order to define " positive multiples " of elements of X, we select an element

xo as a reference point, or origin, in X and start by defining fractional products
ax where x€X and α€[0,1] by the formula ax = (x, x0, a). The following pro-
perties are then valid:

(13) Ox=axo = xo and lx=x.

(14) a(βx) = (aβ)x.
(15) d[ax, ay] = ad[x, y\ in particular if ax=ay and αA=0, then x — y (cancel-
lation law).
(16) β(x,y,a) = (1+^ββ_βx,y,l + aβ-β) if a,βφ,l).

Next we consider in the cartesian product i ? + x l (as before, R+ denotes the

set of positive real numbers), the identification given by

(λ, a?M5, y) if and only if — — x = — — y

λ+δ λ+δ
This is an equivalence relation, to prove the transitivity, for instance, we
notice that if (λ, x)~~(δ, y) and (δ, y)~~(γ, z), i.e.

we get, after appropriate multiplication,

λ δ
y
λ + δ+γ

hence

λ+δ
X z
Γ λ+δ+γ \λ+r )

so that, by cancellation, we have that (λ, x)^(rf z) as wished.

The quotient space (R+xX)/^ will be denoted by C. For the sake of sim-
plicity the equivalence class of the pair (λ, x) and the pair itself will be denoted
in the same way, thus, for instance, we have that (1, xo) = {(λ, xo)t λ>0}.
We now introduce in C an addition, a product by nonnegative scalars and a
metric as follows:
 CONVEX SUBSETS OF NORMED SPACES 311

( i ) δ(λ, x) = (δλ, x) if δ> 0 and 0(λ, x) = (l, xo).

(ii) (λ,x)Hδ,v)=(λ+δ, (x,y,

(in) d'[(λ,x)f (δ,y)] = (

The product as given by (i) is well defined (recall that we are dealing with
classes), furthermore we have that

l(λ, x) — (λ, x) and Λ(l, xQ) = (1, x0).

To see that the addition is well defined, consider the pairs (λ,x), (δ,y) and
(Δ, z) where (λ, x)~(δ, y). As

x— , , V
λ+δ λ+δg
we have after multiplication by (λ+δ)l(λ+δ+Δ), and convex combination with z>
that
/ λ+δ+Δ \ / δ λ+δ+Δ \
)
hence, by (16),
λ+ Δ ( λ \_ δ+Δ I δ \
XZ
λ+δ+2Δ \ ' ' λ + ΔJ" λ+δ+2ά\y'Zt δ+Δj
so that

and, as wanted,

The addition is also commutative and has a zero element, the class (1, x0); to
prove that it is an associative operation, consider the identity in X

λ δ Δ\ ( δ Δ λ\

where s=λ+δ+J. From it we obtain

λ \ λ+δ
V
' ' Ίϊδ ))' Z
'
312 HILTON VIEIRA MACHADO

which, in turn, gives

(λ+δ, (x9 y, - ^ ) ) + ( 4 *) = (*, * ) + ( * + 4 (y, z,

thus,
[«, a?) 4- (3,2/)] + (J, 0) = α x) + [((5, y) + (J, *)]
as we wanted. The two distributive laws are easily verified:
(Δ + Δ')(λ, a?) = Jtf, *) +J'U, ar),

To prove now that d'y as given by (iii), is well defined, notice that if (λ, x)
=(δ,y) we have, after appropriate multiplication, that

thus
Γ λ Δ Ί Γ δ Δ
l δ Δ
X f
z Δ
Z
Γ
d
[
v
' λ+δ+Δ z\
λ λ

hence, by (15),

λ+Δ T λ Δ Ί δ+Δ Γ δ
d Z a
A + J + 3 L A + J ^ λ+A ]"~ λ+δ+Δ lδ+ΔV
multiplication by (λ+δ+Δ) now, leads to
d'[(λ,x), (Δ,z)]=d'l(δ,y), (Δ,z)]
as wanted.
The function d' is seen to satisfy all properties of a metric, only the proof of
the triangle inequality is not entirely trivial: from

Aλ
δ
l^Aλ
Δ
Λ , AΔ
δ
1

where s=λ+δ+Δ, we get, using (15) that

λ+δ J λ δ l^λ +Δ Γ >i J 5 1
' Ί+ϊvl
d d

multiplication by 5 now gives, as wanted,

'l(Δ,z), (δ9y)].
For the sake of reference we now list all properties just proved:
 CONVEX SUBSETS OF NORMED SPACES 313

(17) the addition in C is commutative and associative, there is a zero element

(1, xo) which we will indicate, from now on, by c0.
(18) the product by nonnegative scalars is distributive and associative, we
also have that, if CQC and λ^Q, then

lc=c, λco=Co, Oc=co.

r
(19) d makes C into a metric space.

4. The special character of C.

As a transition between the PF-space X and the final normed space Ef still to
be constructed, the cone C should not only contain a copy of the first, but have,
as well, most characteristics of the second. That this is so, is stablished by the
next three properties:
(20) the metric dr in C is homothetic, i.e. if c and c' belong to C and Λ^O,
then we have that
d'[λc,λc']=M'[c9c'].

(21) df is translation invariant, i.e. for c, c' and c" in C,

c"y c'+c"]=d'[c,c'].

(22) the mapping 77: X-+C given by /7(ar)=(l, x) is an isometry, Π(xQ)=Co

and preserves convex combinations, i.e. for all x and y \τι X and a€[0,1] we
have that
II((x, y, α))=ίt/7(α;) + ( l - α ) % )

Proofs, (20) is proved in a straightforward manner from the definitions.

(21) By condition (C) we have that

td
[{jx'z'T)' {jy'z'j)]=sd[jx'Jy]
where s=λ+δ+d and t=λ+δ+2d. Using now property (16) and (C), on the left
and right members respectively, we have

.Sλ+J/ λ \ δ+Δ( δ \Ί W.ΓJ_ δ Ί

that is,

hence, as wanted,
314 HILTON VIEIRA MACHADO

(22) follows from the fact that

d'[(l,x), (l,y)]=ί

and

for α€(0,1), while that if a=0 or 1 the equality in (22) is trivial.

As direct consequences of (20), (21) and (22) we have that

(23) if c+c" = c' + c", then c=c' (cancellation);

(24) if λc=λc' and λ>0 then c=c';
(25) if λc—λfc and c^Co then λ—λ'\
n
(26) Σ
1

(27) ( U ) = i ( U ) , VΛ>0, #€jζ thus if we identify X and Π(X) we can

write that C=R+X, that is, the cone spanned by X.
Finally we remark that given a metric space X with an operator W: X2 XI-+X,
and a cone C, that is, some other metric space with an algebraic structure satisfy-
ing properties (17) - (21), then the existence of an isometry 77: X-*C preserving
convex combinations as in (22), implies that the operator W satisfies the conditions
(A), (B) and (C), i.e., they are necessary and sufficient conditions for the possibility
of an imbedding as stablished here.

5. The supporting normed space E.

To formalize the construction of the normed space E suggested in section 1

we follow a standard procedure, because of that most proofs will be omitted. We
start by introducing in CxC an equivalence relation: if c,dy e and / are elements
of C we say that

(c,d)—(e,f) if and only if c+f=e+d.

Now we define in the quotient space (CxC)l-, denoted by E, an addition, a

product by scalars and a metric: using the some symbol for the pair (c, d) and the
corresponding equivalence class, we set

(i') (c,d) + (e,f) = (c + e, d+f).

(ii') λ(c,d) = (λc,λd) if Λ^O,
λ(c,d) = \λ\(d,c) if Λ<0.
(iiiO d"[(c,d), (e,f)]=d'[c+f, e+d\.
The addition of classes is well defined, associative and commutative. The
 CONVEX SUBSETS OF NORMED SPACES 315

equivalence class of the pair (cθ9co)t i.e. the set {(c,c), ceQ is a zero element for
the addition and every element (c, d) has a symmetric, namely, (d, c).
The product by real scalars is well defined and satisfies the following identities

(28)
(29) λ[(c, d) + (e, f)]=λ(c, d)+λ(e, / ) .
(30) (λ + δ)(c, d)=λ(c, d)+δ(c, d).
(31) l(c,d) = (c,d).

The mapping d"\ ExE->R is also well defined—it does not depend on the
particular pairs used to represent the equivalence classes. Indeed, if (c, d)^{cu A),
that is c+d!=ci + d, then we have, by property (21), that

d'Ίicudά {e,f)]=d'[Cl+f, e+dj

= d'[c1+f+c+d, e+di+c+d]

=d'[c+f+c1+d, e+d+c+dj

=d'[c+f, e+d]=d"[(c,d), (*,/)].

as wanted. In addition, d" is a metric on E which is translation-invariant and

homothetic, i.e.

), (e,f) + (o,h)]=d"[{c,d), (*,/)].

d"[λ(cfd), λ(e,f)] = \λ\d"[(ctd), (e9f)].
All of these properties, put together, imply that E is a normed vector space
with norm given by the formula

where v=(c,d) is a generic element of E and 0=(co,co) is the zero element.

We now proceed to show the existence of an imbedding of X into the normed
space E as described in Theorem 1. First we consider the mapping C from the
cone C to the space E defined by C(c) = (c,c0). This is easily seen to be an
isometry which preserves addition and multiplication by nonnegative scalars, in
particular C preserves convex combinations and takes the origin c o eC into the
origin of E. The conclusion is that E contains a copy of C, namely the set C(C)\
in fact with this identification in mind we can write that E=C—C in the sense
that any vector v = (c,d) in E can be split as follows:

v=(c, d) = (c, co)-(d, co)= C{c)- C{d).

We are now able to prove Theorem 1, as a matter of fact the proof, after the
preceeding development, is now rather simple.
316 HILTON VIEIRA MACHADO

Proof of Theorem 1. Consider the composition 3' — C °Π of the two mappings

Π: X-*C and C: C->E,

As both, C and 77, are isometries and preserve convex combinations, the same is
true for 3\ the image 3{X) is then a convex subset of the normed space E
which cannot be distinguished from the original TΓ-space X as far as the metric
and convexity structures are concerned.
The point x0, selected as reference in X, gets in this way identified with the
zero vector of the normed space E. After the identification x-+J(x) we can
write that E=R+X-R+X or still, £=span X, indeed if v=(c,d)eE and, say,
c = (λ,x), d=(δ,y) then we have

υ
= (c9d)=c(c)-C(d)=CU(l, x))- C05(1, y))

so that we can think of the normed space E as the minimal extension of X.

6. The uniqueness of the extension.

Theorem 1 stablished that conditions (A), (B) and (C) are necessary and
sufficient for the existence of an imbedding into some normed space. The
uniqueness of the solution of the imbedding problem is guaranteed by the follow-
ing result:

THEOREM 2. Let X be a W-space satisfying (B) and (C). Consider any two
imbeddings J : X->E and 3 \> X-*EX as described in Theorem 1 and assume\ for
simplicity\ that J(#o)=O and Ji(#o)=0i where 0 and Oi are the zero vectors of the
normed spaces E and E\ respectively. Then there is a linear isometry T between
the minimal extensions, span 3(X) and span 3i(X)> which makes the following
diagram commutative

3 span 3{X)aE
X τ\
31 span
+ 4
Proof of Theorem 2. Observing that span 3(X)=R J(X)-R -J(X) we put,
tentatively,

now, if

λJ(x)-δJ(y)=λ'J(x')-δ'J(y')
 CONVEX SUBSETS OF NORMED SPACES 317

we have that
λJ(x)+δ'J(y)=λ'J(x')+δJ(y).
hence, setting s = (λ+δ+λ'+δ')'1, we have

that is
J((x, y', Xo, xo; λs, δ's, λ's, δs)) = J((x', y, x0, xo; λ's, δs, λs, δ's)).

and, as J is 1 — 1,
(x, y', xo, xo\ λs, δ's, λ's, δs) = (x', y, x0, xo; λ's, δs, λs, δ's).

If we now reverse the process, using 3 \ instead of J, we conclude that

proving that T is a well defined mapping from span J(X) to span J ι(X). The
same reasoning shows that T is one-to-one while it is obvious that T is onto.
To prove the additivity of T consider any two points in span J(X), say,
a=λj(x)-δj(y) and a'=λ'J(x')-δ'J(y'); then

so that

If we start with T{a) and T(a') instead, we arrive to the same thing, the conclu-
sion is thus, that T(a + a') = T(a)+T(a'), i.e. that T is additive.
If α>0 we have that aa—aλ J(x) — aδj(y), so that

on the other hand,

while
318 HILTON VIEIRA MACHADO

so that we can conclude that T is homogeneous as well. To see that T is an

isometry consider that

] = d"[λJ(x),

similar development for \\T(a)\\ leads to the same expression so that

as wanted. Finally, if a=J(x)ej(X)9 we have that T(a) = J1(x) so that Γ ° J
= Ji, i.e. the diagram is commutative. This completes the proof of Theorem 2.

7. Final remarks.

We have settled the problem of existence, and uniqueness, of the imbedding

of T^-spaces into normed spaces. It is natural, at this point, to ask whether the
(minimal) extension is necessarily complete, i.e. a Banach space. The answer is,
in general, negative even when the Tf-space is itself a complete metric space, as
the following example shows:

COUNTEREXAMPLE. Consider the subset of I2, defined by

X={xel2, |^|^1/2

clearly X is convex, closed, symmetric and contains the origin, in particular X is

a complete TF-convex metric space. We now prove that the linear subspace span X
(the obvious normed space extension of X) is a proper dense subset of I2. Indeed
it is easy to check that
2 n
spanX=R X={xεl , \2 xn\^M Vn, where M=M(x)}

T h u s the point en = (0, •••,0,1,0, •••) lies in the span X, so that this set is dense in
I2. Consider now the point x = (xn)n where xn = 2~n 2. While it is true that x$Γ%
we have

lim 1 2 ^ 1 = + c o ,
71—»oo

so that x does not belong to span X. This settles the problem posed above.
It should also be remarked the fact (used several times in the proofs) that a
mapping which preserves convex combinations of pairs, will necessarily preserve
multiple convex combinations, i.e. the following is true:
 CONVEX SUBSETS OF NORMED SPACES 319

LEMMA. Let X and Y be two W-spaces the convex combinations in X and

Y will be denoted by (x, y> a) and {x, y, a} respectively. If the mapping f: X—> Y
is such that

), f(v),oc} V
then f preserves multiple combinations, i.e.

f((xi,~ ,Xn, <xi,' ',an)) = {f(x1),~ ,f(xn); <*i, ••-,«»}

for xu --yXn in X and alf •• , α w € / , Σat = l,

Finally we remark the possibility of replacing the system of conditions (A),

(B) and (C) by simpler or geometrically more appealing equivalent systems. Thus,
for instance, condition (A) may be replaced by

(A') (z,z,a) = z, V*€X, α€/.

Property (3), section 2, states that (A') is implied by (A), (B) and (C); we
must now prove that (A'), (B) and (C), together, imply condition (A). To do this
we first observe that for all x, y,

z = (x, y, z\ 0, 0, l) = (y, z, x\ 0,1, 0) = ((y, z, 0), x, 1)

so that

((y, z} 0), x, l) = ((s, z, 0), x, 1) = («, xt 1)

thus, by condition (C),

(y,z,0)=z, V
and, going back, we also conclude that

Using these facts we have, for et^l,

O, y, a) = ((x, y, a), x, 1) = (x, y, x\ a, 1 — α, 0)

= (?/, x, .r; 1-α, 0, a) = ((?/, x, 1), x, l-a) = (y, x,l-a)

while that, for α = l, we have

(x,yfl) = x = (y,x,0)
so that in general,

(x,y,a) = (yfx,l-a).

Now we can prove (A): from the previous considerations we have that
320 HILTON VIEIRA MACHADO

d[z, (x, y, a)]^d[z, (z, y, α)] + d[(2, y} α), {x> yy a)]

= d[(z,z,l — a), (y, z, l-a)] + ad[z, x]

= (1 — a)d[Zy y]+ad[z, x]

which is exactly condition (A).

REFERENCES

[ 1 ] BLUMENTHAL, L. M., Theory and applications of distance geometry. Oxford

(1953).
[ 2] BLUMENTHAL, L. M., AND K. MENGER, Studies in geometry. San Francisco (1970).
[ 3 ] BROWDER, F. E., Nonexpansive nonlinear operators in Banach spaces. Proc. Nat.
Acad. Sci. U.S.A. 54 (1965), 1041-1044.
[4] DUNFORD, N., AND J. T. SCHWARTZ, Linear operators. New York (1955).
[ 5] KIJIMA, YM AND W. TAKAHASHI, A fixed-point theorem for nonexpansive map-
pings in metric space. Kδdai Math. Sem. Rep. 21 (1969), 326-330.
[ 6 ] KIRK, W. A., A fixed-point theorem for mappings which do not increase dis-
tance. Amer. Math. Monthly 72 (1965), 1004-1006.
[ 7 ] MACHADO, H. V., Fixed-point theorems for nonexpansive mappings in metric
spaces with normal structure. Ph. D. thesis, Chicago (1971).
[ 8 ] TAKAHASHI, W., A convexity in metric space and nonexpansive mappings, I.
Kδdai Math. Rep. 22 (1970), 142-149.

DEPARTAMENTO DE MATEMATICA,
UNIVERSIDADE DE BRASILIA.