Nonexpansive selections of metric projections in spaces of continuous functions

NONEXPANSIVE SELECTIONS OF METRIC PROJECTIONSIN SPACES OF BOUNDED FUNCTIONS (DRAFT VERSION)

RAFAEL ESPINOLA AND GENARO LOPEZ

Abstract. The paper is concerned with the study of those subsets of spaces ofcontinuous and bounded functions for which it is possible to find a nonexpan-sive selection of the metric projection. We first characterize those hyperplanesof these spaces with the above property. Later we use these results in order tofind more general subsets with the same property.

1. INTRODUCTION

The aim of this paper is the study of nonexpansive selections of the metric projec-tion onto subsets of spaces of bounded functions. This is a very common propertyin Hilbert spaces since their closed convex subsets enjoy this property. Howeveroutside the Euclidean geometry this property becomes a very exceptional one al-though still very useful and interesting. Metric projections have been extensivelystudied for the last decades and have turned to be a tool of great importance inApproximation Theory and other fields. Metric projections are, in general, mul-tivalued mappings and their properties are basically determined by the geometryof the space. As multivalued mappings they are not that easy to handle and so ithas been the interest of many authors to fix good conditions under which metricprojections admit “nice selections”. Most of these works have a purely linear mo-tivation as are focused on metric projections onto subspaces and they are relatedto the study of norm-one and minimal projections onto finite or co-finite dimen-sional subspaces (see for instance [3, 5, 10, 11, 12, 18]). The reader may also check[13, 14] for different studies concerning metric projections. Our present work hasbeen motivated by previous works from Hsiao and Smarzewski [8], and Ubhaya[19] (see also [20]) where it was shown that it is possible to find nice selections ofthe metric projections onto (non-neccesariarly) linear subsets. In the more generalframework of hyperconvex metric spaces, relevant results in the same direction werebeen found by Sine in [17] and, more recently, by Kirk and the authors in [7].

We begin by introducing basic definitions and preliminary results. Let X denotea metric space and let A ⊆ X nonempty. The metric projection

PA : X → P(A)

of X into the set of subsets of A is defined by

PA(x) = {y ∈ A : d(x, y) = d(x,A)}

1991 Mathematics Subject Classification. Primary 54C20,54E35; Secondary 54C55.Key words and phrases. Spaces of bounded functions, hyperplane, hyperconvexity, metric pro-

jection, selection of multivalued mappings.

1

2 RAFAEL ESPINOLA AND GENARO LOPEZ

where d(x, A) denotes the usual metric distance of x ∈ X from A ⊆ X. WhenPA(x) 6= ∅ we say that A is a proximinal subset of X. A function

f : X → A

is said to be a proximinal retraction onto A if it is a selection of PA, that is, iff(x) ∈ PA(x) for each x ∈ X. f is said to be a proximinal nonexpansive retractiononto A if, in addition,

d(f(x), f(y)) ≤ d(x, y)

for each x and y in X. A will be said to be a proximinal nonexpansive retract of Xif there exists proximinal nonexpansive retraction onto A.

Throughout this work L(I) (just L if no confusion arises) will denote the spaceof (real) bounded functions on an index set I endowed with the supremum norm.If I = N then we will simple denote it by `∞, if I is finite we will write Rn withn = #(I). C(K) will denote the space of all continuous real functions on a compactHausdorff space K.

We will also need definitions and facts from the theory of hyperconvex metricspaces. Hyperconvex spaces were introduced in 1965 by Aronszajn and Panitch-pakdi [1], an example of such spaces is given by any space of bounded functionson an index set, this fact will be used later in this work. Some of the most char-acteristic properties of hyperconvex metric spaces are related to the existence ofextensions of mappings as well as to the existence of nonexpansive retractions (see[6]). A first result regarding proximinal nonexpansive retracts was given by Sine in[17], where he proved that any admissible subset (i.e., intersection of closed balls)of a hyperconvex space X is a proximinal nonexpansive retract of X. This remark-able fact was later extended by Kirk and the authors in [7]. The so-called weaklyexternally hyperconvex sets of a metric space were introduced there. This class ofsets is closely related to the property of being a proximinal nonexpansive retract.In particular Theorem 4.2 in [7] states that a compact subset D of a hyperconvexmetric space X is weakly externally hyperconvex if and only if it is a proximinalnonexpansive retract of X. It is not hard to see that the same result holds forboundedly compact subsets too. The following characterization (Corollary 2.1 in[7]) will be very useful in this work.Theorem 1.1. A subset D of a hyperconvex metric space X is weakly externallyhyperconvex if and only if D is a proximinal nonexpansive retract of D ∪ {z} forany z ∈ X \D.

We now turn to a description of the sections contained in this work. In Section2 we characterize those co-dimension 1 subspaces of spaces of continuous functionswhich are proximinal nonexpansive retracts. Our result improves other resultsalready existing in the literature as Theorem 2.1 in [2]. We show that the existenceof isolated points in the compact Hausdorff set K is crucial for the existence of suchco-dimension 1 subspaces. This fact breaks previous similarities between spaces ofbounded functions and spaces of continuous functions studied by Ubhaya in [19]and [20]. In section 3 we take up the problem of characterizing convex proximinalnonexpansive retracts of spaces of bounded functions. We study the structure ofthese sets in Example 3.1 and Theorem 3.2. In particular it is shown that proximinalnonexpansive retracts of R2 are convex while they need not be convex in spaces oflarger dimension. Then we give different conditions under which convex sets areproximinal nonexpansive retracts. Finally in section 4 we show that the techniques

SELECTIONS OF METRIC PROJECTIONS 3

developed in the previous sections can be used to identify certain subsets of spacesof bounded functions, like the class of certain lipschitzian functions or the class ofincreasing functions on [0, 1], as proximinal nonexpansive retracts.

2. METRIC PROJECTIONS ONTO HYPERPLANES OFSPACES OF CONTINUOUS FUNCTIONS

In this section we focus on the study of hyperplanes of spaces of continuousfunctions. In particular we give necessary and sufficient conditions for a hyperplaneto be a proximinal nonexpansive retract. Notice that our theorem is an extensionof Theorem 2.1 in [2].Theorem 2.1. Let K be a compact Hausdorff space and let H be a proximinalnonexpansive co-dimension 1 subspace of C(K). Then H is a translate of one ofthe following three types:

I. There is an isolated point s0 ∈ K such that H = {f : f(s0) = 0}.II. There are two isolated points s1, s2 ∈ K such that

H = {f : f(s1) = f(s2)}.III. There are two isolated points s1, s2 ∈ K such that

H = {f : f(s1) = −f(s2)}.The converse of this theorem is also true. Indeed, for H in any of the previous

cases is a proximinal nonexpansive subspace:Case I. Rf(s) = f(s) if s 6= s0; Rf(s0) = 0.

Case II. Rf(s) = f(s) if s 6= s1, s2; Rf(s1) = Rf(s2) =f(s1) + f(s2)

2.

Case III. Rf(s) = f(s) if s 6= s1, s2; Rf(s1) = −Rf(s2) =f(s1)− f(s2)

2.

Notice that in all these three cases the retractions are lineal and unique in thesense there is no other way to define a proximinal nonexpansive retraction ontothe above hyperplanes. Notice also that closed half-spaces defined by these co-dimension 1 subspaces are nonexpansive proximinal subsets too.

In the sequel we will denote hyperplanes by linear forms µ ∈ C(K)? as H = {f :µ(f) = 1}.Lemma 2.2. If d(0,H) = 1, then ‖µ‖ = 1.

Proof. Let f ∈ H be a nearest point from H to the origin. Then ‖f‖ = d(0,H) = 1,and ‖µ‖ ≥ µ(f) = 1. If ‖µ‖ > 1 then there exits g ∈ C(K) such that ‖g‖ < 1 andµ(g) = 1, hence g ∈ H and d(0, g) < 1 which contradicts the fact that d(0,H) =1. ¤

We shall consider µ as a measure on K, and assume that R : C(K) → H is aproximinal nonexpansive selection onto H. We also fix F as R(0).Lemma 2.3. Let s be an atom of µ with |µ|(s) = α < 1

2 (the value α = 0 is alsoallowed). Then F (s) = 0.

Proof. Replacing, if necessary, µ by −µ and H by −H, we may assume α ≥ 0. Fixε small enough. Take two disjoint open sets U1 and U2 such that

s ∈ U1, |µ|(U1) < α + ε and |µ|(U2) > 1− α− 2ε.


Now let g1, g2 ∈ C(K) such that ‖g1‖ = ‖g2‖ = 1, gi is supported in Ui, g1(s) = 1and

α + ε >

∫

U1

g1 dµ > α− ε,

∫

U2

g2 dµ > 1− α− 3ε.

Put β+ =∫

(g1 + g2) dµ, β− =∫

(−g1 + g2) dµ, and define f+ =g1 + g2

β+,

f− =−g1 + g2

β−. Note that ‖f±‖ = 1

β±= ±f±(s), and f± ∈ H (notice that ε small

enough makes β− ≥ 0). Since R is a nonexpansive retraction we obtain

|F (s)− f±(s)| ≤ ‖F − f ± ‖ = ‖R(0)−Rf±‖ ≤ ‖f±‖ =1

β±,

but

|F (s)− f±(s)| = |F (s)∓ 1β±|.

Thus |F (s)− 1β+| ≤ 1

β+, so F (s) ≥ 0. But, on the other hand, |F (s) +

1β−| ≤ 1

β−and so F (s) ≤ 0, which completes the proof. ¤

Proof of the Theorem 2.1. Since R is a selection of the metric projection, F = R(0)is a nearest point ro 0 from H, and since d(0,H) = 1 it follows that ‖F‖ = 1. Thus∫

F dµ = 1 implies that |F (s)| ≡ 1 µ-a.e. Since F (s) = 0 for every s ∈ K which isnot an atom of size |α| ≥ 1

2 it follows that µ is concentrated in these atoms, andthere are exactly two possibilities:

1) µ has just one atom of size 1, say µ(s0) = 1. Since F is continuous andF (s) = 0 for s 6= s0, the condition |F (s0)| = 1 implies that so is isolated.Thus we are in case I.

2) µ has two atoms of size 12 , say |µ|(s1) = |µ|(s2) = 1

2 . Again s1, s2 must beisolated and we are in case II or III.

¤

Theorem 2.1 is an extension of Theorem 2.1 in [2]. This extension can be seen intwo different ways, first, Theorem 2.1 in [2] only applies to the space of sequences `∞

and second, it only searches for proximinal nonexpansive and linear retractions ontohyperplanes. It does not seem to be so trivial, however, that our result restrictedto `∞ improves that one from [2]. In fact the following question may be of interest:if there exists a proximinal nonlinear and nonexpansive retraction onto a subspaceof a normed space, does it implies that there must also exist a linear one? Noticethat our result implies a positive answer for hyperplanes of spaces of continuousfunctions. Actually alike questions are not new in the literature, indeed the samequestion has been studied by different authors without the proximality condition.In particular it is known (see Proposition 2.1 in [16] or Section 5.d in [12]) that ifthere exist nonexpansive retractions onto hyperplanes of normed spaces then theremust be at least a linear one. Moreover, if the normed space is a hyperconvexnormed space (i.e., C(E) with E extremely disconnected compact Hausdorff) thenthe above statement applies to subspaces of any dimension or co-dimension. Theo-rem 3.8 below will provide a negative answer to our question at least for subspacesof dimension 1.


3. PROXIMINAL NONEXPANSIVE RETRACTS

This section is devoted to the study proximinal nonexpansive retracts of spacesof bounded function. We begin with an example of a proximinal nonexpansiveretract of R3 which is nonconvex.Example 3.1. Let S ⊆ R3 given by

S = {(1− λ

2,λ

2, 1) : λ ∈ [0, 1]} ∪ {(1,−λ

2, 1− λ

2) : λ ∈ [0, 1]},

i.e., if x = ( 12 , 1

2 , 1), y = (1,− 12 , 1

2 ), and z = (1, 0, 1), then S = S1 ∪S2 where S1, S2

are the segment with respectively endpoints x and z, and y and z. Obviously S isnonconvex. We claim that S is a proximinal nonexpansive retract of R3. To provethis we first need the following properties:

(i) S is a metric segment (i.e., S is isometric to a real interval).(ii) If w ∈ R3 \ S and z1, z2 ∈ PS(w) then seg(z1z2) ⊆ PS(w), where seg(z1z2)

denotes the metric segment in S joining z1 and z2.Property (i) directly follows from the easy-to-check fact that given any z1 ∈ S1

and z2 ∈ S2 then d(z1, z2) = d(z1, z) + d(z, z2). In order to prove (ii) it is enoughto see that if z1 and z2 are such that z1 ∈ S1 and z2 ∈ S2 then z ∈ PS(w), othercases are immediate. Let w = (w1, w2, w3), then

d(w, z1) = max{|1− λ1

2− w1|, |λ1

2− w2|, |1− w3|},

d(w, z2) = max{|1− w1|, | − λ2

2− w2|, |1− λ2

2− w3|}, and

d(w, z) = max{|1− w1|, |w2|, |1− w3|}.Hence it is direct to check that d(w, z) ≤ max{d(w, z1), d(w, z2)} from where (ii)follows.

In order to prove our claim, by Theorem 1.1, it suffices to prove that the metricprojection from S ∪ {w} onto S admits a nonexpansive selection for any w ∈ R3.Let R : S ∪ {w} → S be such that R(w) is the middle point of the metric segmentPS(w), and R|S = Id (where Id stands for the identity map). By definition R is aselection of PS , hence, since R = Id on S, it suffices to prove that

d(R(w), v) ≤ d(w, v) for any v ∈ S.

Now, since PS(w) ⊆ B(w, d(w,S)), if v ∈ PS(w) then

d(R(w), v) ≤ 12length(PS(w)) ≤ d(w, S) = d(w, v)

and hence d(R(w), v) ≤ d(w, v). Let v ∈ S\PS(w) and let u be the closest endpointof PS(w) to v, then we claim that d(v, w) = d(v, u) + d(u,w). Notice that this factimmediately leads to the conclusion since

d(R(w), v) ≤ d(R(w), u) + d(v, u) ≤ d(w, u) + d(v, u) = d(w, v).

The proof of the claim is based on different cases depending on the relative posi-tions of the points v, u and z. Since all these cases follow in a quite straightforwardand similar way we will only outline how to proceed in one of them. Let u, v beas above, both in S1 and in such a way that v is the farthest of them from z, i.e.,v = (1 − λ1

2 , λ12 , 1), u = (1 − λ2

2 , λ22 , 1) and λ1 > λ2. Then a case by case study

implies that d(v, w) and d(u, w) are given by the difference at the same coordinate


which, in addition, must be one of the first two. Suppose it occurs at the firstcoordinate, then f(λ) = |1− λ

2 −w1| must be an increasing function on the interval[λ2, λ1]. Hence 1− λ2

2 < w1 and

d(v, w)− d(u,w) = f(λ1)− f(λ2) =λ1 − λ2

2= d(v, u).

¤After this example we cannot expect that an easy representation of proximinal

nonexpansive retracts is possible, however this does not apply for R2.Theorem 3.2. Let A ⊆ R2 nonempty, then A is a proximinal nonexpansive retractof R2 if and only if it is intersection of closed half-spaces defined by hyperplanes oftype I, II, or III.

Let us first prove some partial results.Lemma 3.3. Let A be a proximinal nonexpansive retract of R2 and let x ∈ A \R2

such that PA(x) contains points of the relative interior of the face C of B(x, d(x,A))supported by the hyperplane H ≡ {x1 = c}. Then H separates x from A.

Proof. Notice first that A is hyperconvex and hence it is metrically convex andconnected (see [6, 7] for details). The following two facts hold:

(i) PA(x) cannot contain points of the interior of two different maximal facesof B(x, d(x,A)).

(ii) C ∩ PA(x) must be either a single point or a (proper) segment.If (i) does not hold then it immediately leads to a contradiction with the metricconvexity of A. If C∩PA(x) is not a singleton then suppose that it is not a segment.Since it is compact there must exist P, Q ∈ C ∩ PA(x) such that

PQ ∩ (C ∩ PA(x)) = {P, Q},where PQ is the segment with endpoints P and Q. Now it is possible to findy ∈ B(x, d(x, A)) such that

PA(y) = {P, Q} and d(y,A) =d(P,Q)

2,

which is a contradiction since no proximinal nonexpansive retraction onto A ishence possible.

So PA(x) is either a point or a segment. Let us first assume it is a segment. Thenit is easy to see that there exists y ∈ B(x, d(x,A)) such that PA(y) = B(y, d(y, A))∩H. Let Q be the middle point of PA(y). Since x and y are in the same half-space defined by H it suffices to prove that H separates A from y. Let R bethe nonexpansive selection of PA, then R(y) = Q. Now, since A is metricallyconvex, if r is the semi-line beginning at Q and passing through y (notice that r isperpendicular to H) then r ∩A = Q. By (i) one of the following facts must hold:

(a) PA(z) ⊆ H for each z ∈ r.(b) There exists z ∈ r such that PA(z) ∩H = ∅.

If PA(z) ⊆ H for each z ∈ r, then a very simple 2-dimensional reasoning leads tothe fact that H separates y from A.

Let us assume that (b) holds. We say that

z > z′ if z, z′ ∈ r and d(z, Q) > d(z′, Q),


then it is clear that there exists z0 ∈ r such that

PA(z) ⊆ H for any z < z0, and

PA(z) ∩H = ∅ for any z > z0.

Obviously z0 ≥ y. Let E and D be the two endpoints of PA(y) and let d = d(E, D).

If z0 = y then we take z > y such that d(z, y) =d

4, then, by (i), we may assume

that D is such that d(D, PA(z)) > d where d(D, z) =3d

4which contradicts the

fact that A is a proximinal nonexpansive retract. Hence only (a) can hold, whichcompletes this part of the proof.

Let us now assume that PA(x) is a singleton in the interior of C. Then there existsy ∈ B(x, d(x,A)) such that PA(y) = PA(x) is an extreme point of B(y, d(y,A)).Let H be the hyperplane of type II or III passing through PA(y) and not throughy, then x lies on the same half-space of those defined by H than y. Notice thatR(y) = PA(y). We claim that H separates y from A. Otherwise, since A isconnected, we may find y0 ∈ seg(y, R(y)) (notice that R(y0) = R(y)) such thatif r is the parallel line to H passing through y0 then r ∩ A 6= ∅. Let z ∈ r ∩ A,then it is a easy exercise to check that d(z, y0) < d(z,R(y0)) which contradicts thenonexpansivity of R. ¤

The next corollary is direct from the last part of the previous proof.Corollary 3.4. If x and A are as in Lemma 3.3 with PA(x) being an extreme pointof B(x, d(x,A)) then there exists a hyperplane of type II or III separating A fromx.

Proof of the Theorem 3.2. After Lemma 3.3 and Corollary 3.4 it only remains toprove the converse implication. We will just indicate how to construct the non-expansive selection of the metric projection, straightforward details will not beincluded. Let us first assume that A is bounded. Since A is an intersection of atmost eight half-spaces, A is the convex hull of a finite collection of points whichwill be the vertexes of A.

If A has nonempty interior then we construct the nonexpansive selection of PA

as follows. Let v be a vertex of A and let H1 and H2 be the two hyperplanesdefining A which contain v. Let r1 and r2 be the half-lines perpendicular to H1

and H2, respectively, and such that ri and A are separated by Hi for i = 1, 2. Werepeat the same operation at each vertex. In this way, if the number of vertex ofA is n (which is to be an integer number in between 3 and 8), R2 \ A is dividedinto 2n different regions, n of which are determined by parallel half-lines and n bynon-parallel ones. If x is in a region of the first type them and r is the line passingthrough x parallel to the borders of the region, then R(x) = r ∩ PA(x). If x is inbetween two non-parallel half-lines then it is enough to define R(x) as the vertexof A which is in the same region than x.

If A is nonbounded but still has vertexes then we work in a similar way. If Ahas no vertex then it is the intersection of (at most) two half-spaces determined byparallel lines. In this case R2 \A is composed of two half-spaces and we just projecteach one onto its closest hyperplane as in Section 2.

If A has empty interior then A must be a (closed) segment sitting in a hyperplaneof those given by the statement. Sine already noticed in Observation (4) of [17]that these segments are proximinal nonexpansive retracts of R2. ¤


As a consequence of this theorem we obtain the following corollary which, asExample 3.1 shows, is no longer true for spaces of larger dimension.Corollary 3.5. If A is a proximinal nonexpansive retract of R2 then A is convex.

After Theorem 3.2 the following two questions are natural:Q 1. Is any convex proximinal nonexpansive retract of a space of bounded func-

tions an intersection of half-spaces as those of Theorem 3.2?Q 2. Does any intersection of half-spaces as above define a proximinal nonex-

pansive retract?The rest of this section is devoted to the study of the above questions. We begin

with a theorem where Question 1 is answered in the negative. The next lemma willbe needed.Lemma 3.6. Let x = (xn) and y = (yn) be two points in `∞ such that |xn| < 1 foreach n ∈ N and ‖(1− λ)x + λy‖ = 1 for each λ ∈ [0, 1], then lim inf |xn − yn| = 0.

Proof. Suppose that |xn−yn| ≥ ε > 0 for n ≥ n0. Thenxn + yn

2< 1 for n ≤ n0 and

xn + yn

2≤ 2− ε

2for n ≥ n0, so, by fixing ε < 2, this leads to a contradiction. ¤

Let us consider the direction v in `∞ given by

v = (1,12− 1,

13− 1, · · · ,

1n− 1, · · · ).

Obviously v is not a direction of any hyperplane of type I, II, or III. Let r beany line of direction v in `∞, then the following corollary follows from Lemma 3.6.Corollary 3.7. Let B be a closed ball of `∞ such that B ∩ r 6= ∅. Then r ∩B is asingleton or contains points of the interior of B.

Proof. There is no loss of generality if we assume that B is the unit ball. Supposethat d(0, r) = 1. We want to see that then B ∩ r is a singleton. Let x ∈ B ∩ r with‖x‖ = 1. Suppose that there is an infinite number of coordinates xnk

of x such that|xnk

| < 1 for any k ∈ N, then if y ∈ B ∩ r and y 6= x we may apply Lemma 3.6to nk’s coordinates and so lim inf |xn − yn| = 0 which contradicts the fact that thevectors x− y and v are dependent.

Now suppose that x, y ∈ B ∩ r are two different points such that |xn| = |yn| = 1for all n ≥ n0 for a certain n0 ∈ N. A very easy study of coordinates n0 andn0 + 1 of x and y shows that the direction x− y must be a direction of one of thehyperplanes of type I, II or III, which again contradicts the fact that x− y and vare dependent. ¤

The following theorem answers Question 1 in the negative.Theorem 3.8. If r is as above (or a translate of it) then it is a proximinal nonex-pansive retract of `∞.

Proof. By Theorem 1.1, it is enough to show that r is a proximinal nonexpansiveretract of r∪{x} for any x ∈ `∞ \ r. Let x ∈ `∞ \ r. There is no loss of generality ifwe assume that x = 0 and d(0, r) = 1. Then, by Corollary 3.7, B ∩ r is a singletonwhich will be denoted as R(0) = (x1, x2, · · · , xn, · · · ). It is only left to prove thatd(R(0), y) ≤ d(0, y) for any y ∈ r.

We claim that:(i) x1 = 1 ⇒ lim sup xn = 1.


(ii) x1 = −1 ⇒ lim inf xn = −1.(iii) −1 < x1 < 1 ⇒ lim supxn = 1 and lim inf xn = −1.

Suppose that x1 = 1 and let a be such that lim sup xn < a < 1. Then there

exists n0 ∈ N such that xn ≤ a for any n ≥ n0. Fix λ =a− 1

2, then

x1 + λv1 =1 + a

2

−1 < −3 + a

4≤ xn + λvn ≤ a + 1

2< 1, for any n ≥ n0.

Now it is possible to choose λ′ ∈ (λ, 0) such that |xn + λ′vn| < 1 for each n ∈{2, 3, · · · , n0}. Hence there exists y ∈ r such that ‖y‖ < 1, which is a contradiction.Statements (ii) and (iii) follow in a similar way. As a consequence one of thefollowing statements holds:

(a) x1 = 1 and lim sup xn = 1.(b) x1 = −1 and lim inf xn = −1.(c) −1 < x1 < 1, lim sup xn = 1 and lim inf xn = −1.

Now, if y ∈ r then y = R(0) + λv for a certain λ, and so

d(R(0), y) = ‖λv‖ = λ,

but, since one of (a), (b) and (c) holds,

d(0, y) = ‖R(0) + λv‖ = 1 + |λ|.Hence R is nonexpansive and the theorem follows. ¤

Remark 3.9. Theorem 3.8 in addition to Theorem 3.5 of [4] gives a negative answerto the question we raised in Section 2 for 1-dimensional subspaces.Remark 3.10. Although Question 1 has been answered in the negative it is clear thatthe infinite-dimensional nature of `∞ is crucial in the definition of r, henceforth itis still natural to ask whether Question 1 remains true for finite dimensional spaces.

Next we give an easy partial answer in the affirmative to Question 1. We willomit its proof.

Theorem 3.11. If A is a proximinal nonexpansive retract of L such that thereexists a collection of hyperplanes {Hα} in such a way that A coincides with theintersection of closed half-spaces determined by these hyperplanes and such thatA ∩ Hα has nonempty interior (relatively to Hα) for each α, then Hα must be ahyperplane of type I, II, or III for each α.

We now take up Question 2. We will not give a definite answer to this questionalthough some partial results in the affirmative are shown. From now on {Hα} willstand for a collection of hyperplanes of L such that Hα is given by an equation oftype I, II, or III, and Hα will stand for the closed half-space defined by Hα. We willsay that i, j ∈ I are directly related by {Hα} if there exists α such that both indexappear in the equation of Hα, we also agree that each index is directly related toitself if it appears in the equation of at least one Hα.

Theorem 3.12. Let A = ∩αHα 6= ∅, then, if for each i ∈ I there is at most onej ∈ I with i 6= j directly related to i, then A is a proximinal nonexpansive retractof L.


Proof. Let y ∈ L \ A, then we define R(y) as follows. If i ∈ I does not appear inany Hα then R(y)(i) = y(i). If the only index related to i is itself then it appearsat most in two hyperplanes given by equations xi = c and xi = d with c ≤ d. Thenwe define

R(y)(i) =

c, if y(i) ≤ c;y(i), if c ≤ y(i) ≤ d;d, if d ≤ y(i).

Now, if i is related to j 6= i, then, by our hypothesis, we may isolate both indexesand work with them as if we were in R2 and so it is enough to define R(y)(i)and R(y)(j) as in Theorem 3.2. Now it is straightforward to check that R is anonexpansive selection of PA. ¤

Let {Hα} be such that A = ∩αHα 6= ∅. We will say that i, j ∈ I are related by{Hα} if we may introduce a hyperplane Hβ given by xi + xj = c or xi − xj = c insuch a way that

∩αHα ∩Hβ = A.

Obviously this relation establishes an equivalence relation on the index set I. Wewill finish this section by proving that such an A is proximinal nonexpansive retract.First we need the following lemma.Lemma 3.13. Let v be a direction of L which i-coordinate is equal to 1 or −1 foreach i ∈ I, then any line r with direction v is a proximinal nonexpansive retract ofL.

Proof. If each coordinate of v is 1 then r is a set as those described by Ubhaya in[20]. Otherwise it is enough to apply isometries to L by properly multiplying thecoordinates of v by −1 so −1’s of v turn into 1’s. Notice also that there is only oneway to define the proximinal nonexpansive retraction onto r. ¤Theorem 3.14. Let A = ∩αHα 6= ∅ as above, then A is a proximinal nonexpansiveretract of L.

Proof. Let y ∈ L \ A, we define each coordinate of R(y) in the following way. If idoes not appear in the equation of any Hα then R(y)(i) = y(i). If there exists αsuch that i appears in Hα then let J ⊂ I be the equivalent class of i in I. Then itis not hard to see that A ∩ L(J) is either a singleton or a line. If A ∩ L(J) = {z}then we make R(y)(j) = z(j) for j ∈ J . If A∩L(J) is a line r then it must be a lineas those described in Lemma 3.13 and hence we define R(y) so as its restriction toJ coincides with the selection of the metric projection from L(J) onto r. ¤

4. Isotone and Lipschitzian functions

In this section we show that Examples 4.3 and 4.4 from [20] can be described asintersections of hyperplanes and half-spaces of type II (it is not hard to see fromTheorem 3.2 in [19] that such intersections are proximinal nonexpansive retracts).

4.1. Isotone functions. Let L = L(I) and Π ⊆ 2I such that ∅, I ∈ Π, and Π isclosed under arbitrary intersections. Then a function x ∈ L(I) is called a (weak)Π-isotone function if {i ∈ I : x(i) > k} ∈ Π for all real k. If, in addition, Π is closedunder arbitrary unions then such functions are called Π-isotone. Ubhaya shows inExample 4.3 of [20] that Π-isotone functions is a proximinal nonexpansive retractof L(I). An example of such functions is given by the set of increasing functions


on the real interval [0, 1], indeed increasing functions on [0, 1] are Π-isotone withrespect to Π = {[x, 1], (x, 1] : x ∈ I}.

In this sub-section we show that the collections of Π-isotone functions are inter-section of hyperplanes and half-spaces determined by equations like xi−xj = 0 fori, j ∈ I.

Let Π ⊆ 2I as above, then for each i ∈ I we set [i] = {A ∈ Π : i ∈ A} andAi = ∩A∈[i]A, by the properties of Π, Ai ∈ Π and Ai 6= ∅ for each i ∈ I. Noticethat in this way we may say that i ≤ j if Ai ⊇ Aj . Then we define LΠ in thefollowing way:

LΠ = ∩{Hi : I ∈ I}, where Hi = ∩j∈Ij≥i

Hji ,

and

Hji =

{xi − xj = 0, if Ai = Aj

xi ≤ xj , if Ai ⊇ Aj .

The following theorem is immediate.

Theorem 4.1. A function x ∈ L(I) is Π-isotone if and only if x ∈ Lπ.

4.2. Lipschitzian functions. In Example 4.4 in [20] the author considers a realcompact interval S and the corresponding space of continuous functions C(S). Thenhe takes

K = {x ∈ C(S) : |x(s)− x(t)| ≤ M |s− t|α},for some fixed M and α, and shows that K is a proximal nonexpansive retract ofC(S) and of L(S). However K may also be described in the following way:

K =⋂

s,t∈S

(H+s,t ∩H−

s,t),

where

H+s,t ≡ x(s)− x(t) ≤ M |s− t|α, and H−

s,t ≡ x(s)− x(t) ≥ −M |s− t|α.

Furthermore, if δ : [0, diam(S)] → R+ is such that limε→0+ δ(ε) = 0, i.e. it is aso-called modulus of continuity, then K ⊆ C(S) given by

Kδ = {x ∈ C(S) : |x(s)− x(t)| ≤ δ(|s− t|)},

can be expressed as

Kδ =⋂

s,t∈S

(H+s,t ∩H−

s,t),

where

H+s,t ≡ x(s)− x(t) ≤ δ(|s− t|), and H−

s,t ≡ x(s)− x(t) ≥ −δ(|s− t|).

Hence Kδ is another example of a proximinal nonexpansive retract of C(S) -as wellas of L(S)- which can be described by hyperplanes and half-spaces of type II.


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Nonexpansive selections of metric projections in spaces of continuous functions

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