Porosity of Convex Nowhere Dense Subsets of Normed Linear Spacesdownloads.hindawi.com/journals/aaa/2009/243604.pdf · 2019. 7. 31. · Normed Linear Spaces Filip Strobin1,2 ... 4

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2009, Article ID 243604, 11 pagesdoi:10.1155/2009/243604

Research ArticlePorosity of Convex Nowhere Dense Subsets ofNormed Linear Spaces

Filip Strobin1, 2

1 Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, P.O. Box 21,00-956 Warszawa, Poland

2 Institute of Mathematics of the Technical University of Łodz, Wolczanska 215,93-005, Łodz, Poland

Correspondence should be addressed to Filip Strobin, [email protected]

Received 9 June 2009; Revised 24 September 2009; Accepted 29 November 2009

Recommended by Simeon Reich

This paper is devoted to the following question: how to characterize convex nowhere dense subsetsof normed linear spaces in terms of porosity? The motivation for this study originates from papersof V. Olevskii and L. Zajıcek, where it is shown that convex nowhere dense subsets of normedlinear spaces are porous in some strong senses.

Copyright q 2009 Filip Strobin. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

1. Introduction

The paper concerns the topic of describing smallness of interesting sets of metric spacesin terms of porosity. The notions of porosity and σ-porosity (a set is σ-porous if it is acountable union of porous sets) can be considered as stronger versions of nowhere densityand meagerness—in particular, in any “reasonable” metric space, there exist sets which arenowhere dense and are not σ-porous. Thus it is interesting to know that some sets are not onlynowhere dense (meager) but even porous (σ-porous). In such a direction many earlier resultswere extended, for example, it turned out that the set of all Banach contractions was not onlymeager but also σ-porous in the space of all nonexpansive mappings (cf. [1, 2]). Since thereare various types of porosity (more or less restrictive), the natural problem of finding themost restrictive notion of porosity, which would be suitable for an examined set, is also aninteresting task. The reader who is not familiar with porosity is referred to the survey papers[3, 4] on porosity on the real line, metric spaces, and normed linear spaces.

In the paper we try to answer the following question: what is the best approximationof smallness (in terms of porosity) of convex nowhere dense subsets of normed linear spaces?Zajıcek [4] observed that such sets are R-ball porous for every R > 0, and 0-cone porous (cf.[4, page 518]). In fact, Zajıcek’s observation is an improvement of the earlier result of Olevskii

2 Abstract and Applied Analysis

[5] (as was shown in [6], Olevskii worked with a much weaker version of porosity than R-ball porosity). Hence for our purpose we need to find some stronger condition, which wouldimply R-ball porosity for every R > 0, and 0-cone porosity.

The paper is organized as follows. In Section 2, we give definitions of some types ofporosity, that is, R-ball porosity, 0-angle porosity (a stronger version of 0-cone porosity) andintroduce the notion of c-porosity. We also make some basic observations (i.e., c-porosity⇒ 0-angle porosity ⇒ R-ball porosity) and demonstrate that c-porosity gives the characterizationof smallness of convex nowhere dense sets.

In Section 3, we prove that in any Hilbert space H with dimH > 1, the unit sphereis 0-angle porous and is not a countable union of c-porous sets (i.e., is not σ-c-porous). Thisobservation shows that the notion of 0-angle porosity is quite far from the notion of c-porosity.

The motivation for Section 4 originates from the fact that the notion of c-porosity usesthe space of all continuous functionalsX∗. In this section we discuss the possibility of findingthe best approximation of smallness of a convex nowhere dense sets in terms of porositywithout using X∗.

In Section 5, we give one example of σ-c-porous subset of the space of continuousfunctions. For other interesting σ-c-porous sets, we refer the reader to [5] (one of them dealswith the Banach-Steinhaus principle).

2. Some Notions of Porosity

In this section we present the definitions of R-ball porosity, ball smallness, (σ-)0-angleporosity, and (σ-)c-porosity. We also make some basic observations, which will be used inthe sequel (see Proposition 2.8 and Example 2.9).

Let (X, ‖ · ‖) be a real normed linear space and M ⊂ X. Given x ∈ X and r > 0, wedenote by B(x, r) the open ball with center x and radius r. By X∗ we denote the space of allcontinuous linear functionals on X.

Definition 2.1 (see [4, 7]). Let R > 0. We say that M is R-ball porous if for any x ∈ M andα ∈ (0, 1), there exists y ∈ X such that ‖x − y‖ = R and B(y, αR) ∩M = ∅.

Remark 2.2. The definition ofR-ball porosity presented in [7], [4, page 516] is slightly differentfrom the above one. Namely, M is R-ball porous if for any x ∈ M and ε ∈ (0, R), there existsy ∈ X such that ‖x − y‖ = R and B(y,R − ε) ∩ M = ∅. However, it is obvious that bothdefinitions are equivalent.

Definition 2.3 (see [4, 7]). We say that M is 0-angle porous if for every x ∈ M and every r > 0,there exist y ∈ B(x, r) and φ ∈ X∗ \ {0}such that

{z ∈ X : φ(z) > φ

(y)} ∩M = ∅. (2.1)

Note that 0-angle porosity can be considered as a “global” version of (mentioned inthe introduction) 0-cone porosity and, in particular, 0-angle porosity implies 0-cone porosity.

For the definitions of α-cone porosity and α-angle porosity, where α ∈ [0, 1), see [4,page 516] and [7], respectively.

Abstract and Applied Analysis 3

Definition 2.4. M is called c-porous if for any x ∈ X and every r > 0, there are y ∈ B(x, r) andφ ∈ X∗ \ {0} such that

{z ∈ X : φ(z) > φ

(y)} ∩M = ∅. (2.2)

C-porosity turns out to be the suitable notion to describe the smallness of convexnowhere dense sets (see Proposition 2.5) and is a stronger form of 0-angle porosity (x ∈ Xinstead x ∈ M). Indeed, consider the unit sphere S of any nontrivial normed space. S is notc-porous (simply take x = 0 and r = 1/2) and is 0-angle porous—to see it, use the Hahn-Banach separation theorem (cf. [8]) for sets B(0, 1) (the closure of B(0, 1)) and {(1 + r/2)y},where y ∈ S and r > 0.

If a set M is a countable union of c-porous sets, then we say that M is σ-c-porous. Inthe same way we define σ- 0-angle porosity. If M =

⋃n∈N

Mn and each Mn is Rn-ball porousfor some Rn > 0, then we say that M is ball small.

The next result shows that c-porosity is the best approximation of smallness (in thesense of porosity) of convex nowhere dense sets (in the proof we extend an argumentsuggested by Zajıcek [4, page 518]).

Proposition 2.5. A subset M of a normed space X is c-porous if and only if conv M is nowheredense.

Proof. “⇒” It is obvious that for any φ ∈ X∗ and y ∈ X, we have

{z : φ(z) > φ

(y)} ∩M = ∅ ⇐⇒ {

z : φ(z) > φ(y)} ∩ convM = ∅. (2.3)

Hence ifM is c-porous, then convM is also c-porous and, in particular, nowhere dense.“⇐” Fix any x ∈ X and r > 0. Since convM (the closure of convM) is nowhere dense,

there exists y ∈ B(x, r) \ convM. Sets convM and {y} satisfy the assumptions of the Hahn-Banach separation theorem, so there exist φ ∈ X∗ and c ∈ R such that φ(y) > c and for anyz ∈ convM,φ(z) < c. ThenM ∩ {z : φ(z) > φ(y)} = ∅.

Corollary 2.6. Let X be any normed space and let (∗) be any condition such that

if A ⊂ X satisfies (∗) and B ⊂ A, then B satisfies (∗). (2.4)

If every convex and nowhere dense subset ofX satisfies (∗), then any c-porous subset ofX satisfies (∗).

The notions of 0-angle porosity and c-porosity involve the space X∗; however, in itsorigin the porosity was defined in metric spaces. In the next part of this section we will showwhat kind of porosity without using X∗ is implied by them (see Proposition 2.8). Note thatwe will use this result in Sections 3 and 4.

We omit the proof of the following result since it is technical and can be easily deducedfrom the proof of [5, Lemma 1].


Lemma 2.7. Let R > 0, r ∈ (0, 1/2), x0, y0 ∈ X, φ ∈ X∗ \ {0}. If ‖y0 − x0‖ < rR, then there existsy ∈ X such that ‖y − x0‖ = R and

B(y, (1 − 2r)R

) ⊂ {x ∈ X : φ(x) > φ

(y0)}

. (2.5)

Proposition 2.8. The following statements hold.

(i) IfM is 0-angle porous, thenM is R-ball porous for every R > 0, that is,

for every R > 0, x ∈ M and every α ∈ (0, 1),

there exists y ∈ X with∥∥y − x

∥∥ = R and B

(y, αR

) ∩M = ∅.(2.6)

(ii) IfM is c-porous, then

for every R > 0, x ∈ X and every α ∈ (0, 1),

there exists y ∈ X with∥∥y − x

∥∥ = R and B(y, αR

) ∩M = ∅.(2.7)

Proof. We will prove only (i), since the proof of (ii) is very similar. Fix R > 0, x0 ∈ M andα ∈ (0, 1). Let r > 0 be such that 1 − 2r > α, and let y0 ∈ X and φ ∈ X∗ \ {0} be such that‖y0 −x0‖ < rR andM∩{x : φ(x) > φ(y0)} = ∅. By Lemma 2.7, we have that there exists y ∈ Xsuch that ‖y−x0‖ = R andB(y, (1−2r)R) ⊂ {x : φ(x) > φ(y0)}. SinceB(y, αR) ⊂ B(y, (1−2r)R),the result follows.

Note that (2.7) is stronger than (2.6). Indeed, the unit sphere in any normed spacesatisfies (2.6) and does not satisfy (2.7). In the sequel, we will extend this observation (seeTheorem 3.2).

The next example shows, in particular, that the converse of the Proposition 2.8 is nottrue.

Example 2.9. Let (X, ‖ · ‖) be one of the following real Banach spaces: c0 or lp, p ∈ [1,∞). Letus define the set M :=

⋃n∈N

{±ne1,±ne2, . . . ,±nen}, where

en(r) :=

⎧⎨

⎩

1, r = n,

0, r /=n.(2.8)

Now we will show that M satisfies the following condition, which is stronger than(2.7) (and, in particular, than (2.6)):

for every R > 0, x ∈ X, there exists y ∈ X s.t.∥∥y − x

∥∥ = R and B(y,R

) ∩M = ∅. (2.9)


To see it, take any x ∈ X and R > 0. Since x(n) → 0, there exists n0 > 3R such that|x(n0)| < R. Assume, without loss of generality, that x(n0) ≥ 0. Now let y ∈ X be such that

y(k) :=

⎧⎨

⎩

x(k), k /=n0,

x(k) + R, k = n0.(2.10)

Then ‖x − y‖ = R. To see that B(y,R) ∩M = ∅, take any z ∈ M and consider three cases.

(i) z(n0) = 0, then ‖y − z‖ ≥ |x(n0) + R − z(n0)| = |x(n0) + R| ≥ R.

(ii) z(n0) ≥ n0, then ‖y − z‖ ≥ |x(n0) + R − z(n0)| ≥ n0 − x(n0) − R ≥ R.

(iii) z(n0) ≤ −n0, then ‖y − z‖ ≥ |x(n0) + R − z(n0)| ≥ n0 + x(n0) + R ≥ R.

Now we will show that M is not 0-angle porous. It is sufficient to show that for anyφ ∈ X∗ \ {0} and y ∈ X, M ∩ {x : φ(x) > φ(y)}/= ∅. Fix any φ ∈ X∗ \ {0} and y ∈ X, thenthere exists a sequence x0 such that φ(x) =

∑∞n=1 x(n)x0(n) for any x ∈ X. Let n1 ∈ N be such

that x0(n1)/= 0. Assume, without loss of generality, that x0(n1) > 0. Let n2 ≥ n1 be such thatn2x0(n1) > φ(y). Then n2en1 ∈ M and

φ(n2en1) = n2x0(n1) > φ(y). (2.11)

Thus M is not 0-angle porous, and hence not c-porous.

3. On c-Porosity

In this section we will show that c-porosity is a much stronger notion of porosity than 0-angleporosity. This will justify introducing this notion.

From now on, if (H, (· | ·)0) is a real Hilbert space, then (R ×H, (· | ·)) denotes the realHilbert space R ×H with the inner product (· | ·) defined as follows:

((a, x) | (b, y)) := ab +

(x | y)0 for any (a, x),

(b, y

) ∈ R ×H. (3.1)

Denote by ‖ · ‖0 and ‖ · ‖ the norms generated by (· | ·)0 and (· | ·), respectively.We will show that in any nontrivial real Hilbert space H with dimH > 1, the unit

sphere S is not σ-c-porous. In fact, we will obtain a more general result. If S =⋃

n∈NSn, then

there exists n0 ∈ N such that Sn0 does not satisfy (2.7), and hence (by Proposition 2.8) S is notσ-c-porous.

Lemma 3.1. Let (H, (· | ·)0) be a nontrivial real Hilbert space. For any δ ∈ (0, 1), the set

Vδ :={(

a, y) ∈ R ×H : 1 − δ ≤ a ≤ 1 and

∥∥(a, y)∥∥ = 1

}(3.2)

does not satisfy (2.7).


Proof. Take (1 − (1/2)δ, 0) ∈ R ×H, R = 3 and

α :=1Rmax

⎧⎨

⎩

√

R2 − 4(√

2δ − δ2 − 12δ +

116

δ2),

√

R2 − 14δ2

⎫⎬

⎭. (3.3)

It is easy to see that α ∈ (0, 1). Let (a0, y0) ∈ R ×H be such that

∥∥∥∥(a0, y0

) −(1 − 1

2δ, 0

)∥∥∥∥ = R. (3.4)

We will show that B((a0, y0), αR) ∩ Vδ /= ∅. Consider the following three cases.

Case 1 (a0 ≤ 1 and y0 /= 0). Then

a0 < −1 or∥∥y0

∥∥0 > 2. (3.5)

Indeed, otherwise we would have a contradiction since

R =∥∥∥∥(a0, y0

) −(1 − 1

2δ, 0

)∥∥∥∥ =

√(a0 −

(1 − 1

2δ

))2

+∥∥y0

∥∥20

=

√

a20 − 2a0

(1 − 1

2δ

)+(1 − 1

2δ

)2

+∥∥y0

∥∥20 ≤

√1 + 2 + 1 + 4 < 3 = R.

(3.6)

Set η :=√2δ − δ2/‖y0‖0. It is easy to see that (1−δ, ηy0) ∈ Vδ. We will show that (1−δ, ηy0) ∈

B((a0, y0), αR). By (3.4), we have

∥∥(1 − δ, ηy0) − (

a0, y0)∥∥2 =

(−12δ −

(a0 −

(1 − 1

2δ

)))2

+∥∥(η − 1

)y0∥∥20

=14δ2 + δ

(a0 −

(1 − 1

2δ

))+(a0 −

(1 − 1

2δ

))2

+ 2δ − δ2 − 2√2δ − δ2

∥∥y0∥∥0 +

∥∥y0∥∥20

= R2 − 14δ2 + (1 + a0)δ − 2

√2δ − δ2‖y0‖0,

(3.7)

so if a0 < −1, then, by (3.3), we infer

∥∥(1 − δ, ηy0) − (

a0, y0)∥∥2

< R2 − 14δ2 ≤ α2R2, (3.8)


and if ‖y0‖0 > 2, then, again by (3.3), we get

∥∥(1 − δ, ηy0

) − (a0, y0

)∥∥2< R2 − 1

4δ2 + 2δ − 4

√2δ − δ2 ≤ α2R2. (3.9)

Case 2 (a0 ≤ 1 and y0 = 0). In this case a0 = (1 − (1/2)δ) − R. Set z ∈ H with ‖z‖0 =√2δ − δ2.

It is obvious that (1 − δ, z) ∈ Vδ. We will show that (1 − δ, z) ∈ B((a0, 0), αR). By (3.3), (3.4),and a fact that R = 3, we get

‖(1 − δ, z) − (a0, 0)‖2 =(1 − δ − 1 +

12δ + R

)2

+ 2δ − δ2

= R2 − δR +14δ2 + 2δ − δ2

= R2 − δ − 34δ2 < R2 − 3

4δ2 < α2R2.

(3.10)

Case 3 (a0 > 1). Take (1, 0) ∈ Vδ. By (3.3) and (3.4) we infer

∥∥(a0, y0) − (1, 0)

∥∥2 =∥∥∥∥

((a0 −

(1 − 1

2δ

))− 12δ, y0

)∥∥∥∥

2

=(a0 −

(1 − 1

2δ

))2

− δ

(a0 −

(1 − 1

2δ

))+14δ2 +

∥∥y0∥∥20

= R2 − δ(a0 − 1) − 14δ2 < R2 − 1

4δ2 ≤ α2R2,

(3.11)

so (1, 0) ∈ Vδ ∩ B((1 − (1/2)δ, 0), αR).As a consequence, in all cases we have B((a0, y0), αR) ∩ Vδ /= ∅, and hence the result

follows.

Theorem 3.2. Let (H, (· | ·)) be any Hilbert space with dimH > 1 and let S be the unit sphere inH. If S =

⋃n∈N

Sn, then there is n0 ∈ N such that Sn0 does not satisfy (2.7). In particular, S is notσ-c-porous.

Proof. The second statement follows from the first one by Proposition 2.8. We will prove thefirst statement. Let (H, (· | ·)0) be a Hilbert space with dimH > 1. Since S is complete, by theBaire Category theorem, there exists n0 ∈ N such that Sn0 is not nowhere dense in S. Hence

there exists a nonempty set U open in S such that U ⊂ Sn0

S= Sn0 (by Sn0

Swe denote the

closure of Sn0 in the space S). Since the closure of a set which satisfies (2.7), also satisfies(2.7), the proof will be completed if we show that U does not satisfy (2.7). Take any x0 ∈ Uand consider one-dimensional subspace M := {ax0 : a ∈ R}. It is well known (see, e.g., [8])that

M⊥ :={y ∈ H :

(y | x0

)0 = 0

}(3.12)


is a closed subspace ofH andH = M ⊕M⊥. Consider the space R ×M⊥. It is easy to see that

the function H � ax0 + yF�→ (a, y) is an isometrical isomorphism between H and R × M⊥.

Since (2.7) is a metric condition, it suffices to show that the set

V := F(U) ={(

a, y) ∈ R ×M⊥ : ax0 + y ∈ U

}(3.13)

does not satisfy (2.7) in R ×M⊥. Since S1 := F(S) = {(a, y) : ‖(a, y)‖ = 1} and F|S : S → S1 isa homeomorphism between S and S1, the set V is open in S1. Hence and by the fact that thepoint (1, 0) is in V, we infer there exists 0 < δ < 1, such that

Vδ ⊂ V, (3.14)

where Vδ is defined as in Lemma 3.1. Indeed, since (1, 0) ∈ V and V is open in S1, we havethat there are c, d ∈ R and r > 0 such that 0 < c < 1 < d and

[(c, d) × B(0, r)] ∩ S1 ⊂ V. (3.15)

Set δ := min{r2/2, (1 − c)/2} and take any (a, y) ∈ Vδ, then

d > 1 ≥ a ≥ 1 − δ > 1 − 1 + c = c,

∥∥y∥∥0 =

√1 − a2 ≤

√1 − (1 − δ)2 =

√2δ − δ2 <

√2δ ≤ r,

(3.16)

so (a, y) ∈ [(c, d) × B(0, r)] ∩ S1 which yields (3.14). Since Vδ does not satisfy (2.7) in view ofLemma 3.1, the proof is completed.

Now we show that for the Euclidean space R, all presented notions of porositycoincide. In [7, page 222] it is given that any ball small subset of R is countable. Thus and byProposition 2.8, if M ⊂ R, then M is σ-c-porous ⇔ M is a countable union of sets satisfying(2.7) ⇔ M is σ-0-angle porous ⇔ M is ball small ⇔ M is countable.

4. Smallness of Convex Nowhere Dense Sets in Terms ofPorosity without Using X∗

In this section we will discuss the problem of finding the best approximation of smallnessof a convex nowhere dense subset of a normed space X in terms of porosity without usingX∗ (as was mentioned, in its origin porosity was defined as a strictly metric condition). ByPropositions 2.5 and 2.8, any such set satisfies (2.7). This is a stronger version of the first partof Zajıcek’s observation, which states that such sets are R-ball porous for every R > 0. Indeed,by Theorem 3.2, the unit sphere in Hilbert space is R-ball porous for every R > 0 and is not acountable union of sets satisfying (2.7).

Now let M be the set defined in Example 2.9. M satisfies (2.9), hence (2.7), and is not0-angle porous, so is not c-porous. This shows that, in general, the notion of c-porosity ismore restrictive than condition (2.7). On the other hand, as was mentioned, in any nontrivial


normed linear space, the unit sphere (which is 0-angle porous) does not satisfy (2.7), andhence, in general, the notion of 0-angle porosity and condition (2.7) are not comparable.

Clearly, condition (2.7) is only one of possible stronger versions of R-ball porosity forevery R > 0. The other are condition (2.9) and the following weakening of (2.9):

for any R > 0 and any x ∈ M, there exists y ∈ X

such that(‖x − y‖ = R and B

(y,R

) ∩M = ∅).(4.1)

Now the question arises whether any convex nowhere dense subset of any normedlinear space satisfies (4.1) or (2.9)?

Since the closed balls in finite dimensional normed spaces are compact, conditions(2.9) and (2.7) are equivalent in such spaces (note that a similar result is given in [9, Remark2.4]), and hence any convex nowhere dense subset of such space satisfies (2.9). However, inthe remainder of this section we will show that in a very wide class of Banach spaces there aresets, which are convex and nowhere dense, and are not a countable union of sets satisfying(4.1).

Let us focus our attention on nonreflexive spaces.

Proposition 4.1. Let (X, ‖ ·‖) be a real nonreflexive Banach space. Then there exists a closed subspaceM � X, which is not a countable union of sets satisfying (4.1).

Proof. SinceX is a nonreflexive Banach space, there exists a closed subspaceM � X such thatfor every x0 ∈ M,R > 0 and every y ∈ X, if ‖y − x0‖ = R, then B(y,R) ∩M/= ∅ (this is a wellknown fact which follows from the James’ theorem [10, page 52]). We will show thatM is nota countable union of sets satisfying (4.1). Assume that M =

⋃n∈N

Mn. Since M is complete,by the Baire Category theorem, there exists n0 ∈ N such thatMn0 is not nowhere dense inM.Hence for some x ∈ M and r > 0, we have that

B(x, r) ∩M ⊂ Mn0

M= Mn0 . (4.2)

Since x ∈ Mn0 , there exist x0 ∈ Mn0 and r1 > 0 such that B(x0, r1) ⊂ B(x, r). Then

B(x0, r1) ∩M ⊂ Mn0 . (4.3)

Fix any R > 0 and let y ∈ X be such that ‖y − x0‖ = R. Then there exists z ∈ B(y,R) ∩ M,and then the segment (x0, z] ⊂ B(y,R) ∩ M. Hence if x′ ∈ (x0, z] is such that ‖x0 − x′‖ <

r1, then x′ ∈ B(y,R) ∩ B(x0, r1) ∩ M ⊂ B(y,R) ∩ Mn0 . Thus B(y,R) ∩ Mn0 /= ∅, and henceB(y,R) ∩Mn0 /= ∅.

A natural question arises, what happens in reflexive spaces?

Example 4.2. An anonymous referee observed that the Hilbert cube

K ={x ∈ l2 : − 1

n≤ x(n) ≤ 1

n, n ∈ N

}(4.4)


does not satisfy (4.1). To see it, recall the concept of the so-called supported points. We sayx ∈ M ⊂ X is a supported point ofM, if there exists φ ∈ X∗ \ {0} such that φ(x) = sup{φ(y) :y ∈ M}; if such a functional does not exist, then x is called a nonsupported point (cf. [11,page 44]). Now take x0 = 0 and R = 1. Then it is easy to see that x0 is a nonsupported pointofK. Now assume that y ∈ l2 is such that ‖y − x0‖ = ‖y‖ = 1 and B(y, 1) ∩K = ∅. Then by theHahn-Banach separation theorem (cf. [8, page 38]), there exists φ ∈ l∗2 \ {0}with

sup{φ(z) : z ∈ K

} ≤ inf{φ(z) : z ∈ B

(y, 1

)}. (4.5)

On the other hand, x0 is on the boundary of B(y, 1), and hence

φ(x0) ≥ inf{φ(z) : z ∈ B

(y, 1

)}. (4.6)

This gives a contradiction. Hence K does not satisfy (4.1).

By Proposition 4.1 and the previous example, condition (2.7) seems to be quite suitablefor describing smallness of convex nowhere dense sets in terms of porosity without usingX∗.However, the next example shows that there are sets which satisfy (4.1) (and, in particular,(2.7)) and are not a countable union of c-porous sets.

Example 4.3. Let X = R2, ‖(x, y)‖ =| x | +|y| and let

M ={(x, y

): x ≥ 1√

2and x2 + y2 = 1

}. (4.7)

It is easy to see that M satisfies (4.1). Moreover,using an analogous method as in the proofof Lemma 2.7, it can be easily shown that M is not σ-c-porous.

5. Applications

We will give one example of σ-c-porous set (for other, we refer the reader to [5]).LetH be a Hilbert space, and letK be a nonempty bounded closed and convex subset

of H. Define CB(K) := {A : K → H : A is continuous and A(K) is bounded in H}. ConsiderCB(K) as a Banach space with the norm ‖A‖ := supx∈K‖A(x)‖. Let kB be the set of all Banachcontractions:

kB ={A : K −→ K : ∃α∈(0,1)∀x,y∈K

∥∥Ax −Ay∥∥ ≤ α

∥∥x − y∥∥}. (5.1)

For any α ∈ (0, 1), we also define kBα := {A : K → K : ∀x,y∈K‖Ax −Ay‖ ≤ α‖x − y‖}.

Proposition 5.1. kB is a σ-c-porous subset of CB(K). In particular, kB is ball small.

Proof. De Blasi and Myjak [1] (cf. also [2]) proved that for any α < 1, kBα is lower porous(and hence nowhere dense; for the definition of lower porosity, see [4]) subset of the space

Ω :={A : K −→ K : ∀x,y∈K

∥∥Ax −Ay∥∥ ≤ ∥∥x − y

∥∥}, (5.2)


with the metric induced from (CB(K), ‖ · ‖). Hence kBα is a nowhere dense subset of CB(K).It is also obvious that kBα is convex. As a consequence, for any α < 1, the set kBα is a c-poroussubset of CB(K). Since kB =

⋃n∈N

kB(1−1/n), the proof is completed.

Acknowledgments

The author is very grateful to the anonymous referees and Simeon Reich for suggestionshow to modify the paper, and for solving the problem concerning condition (4.1) in reflexivespaces. Also, he would like to thank Szymon Głab and Jacek Jachymski for many valuablediscussions.

References

[1] F. S. De Blasi and J. Myjak, “Sur la porosite de l’ensemble des contractions sans point fixe,” ComptesRendus de l’Academie des Sciences Paris, vol. 308, no. 2, pp. 51–54, 1989.

[2] S. Reich, “Genericity and porosity in nonlinear analysis and optimization,” in Proceedings of theComputer Methods and Systems (CMS ’05), pp. 9–15, Cracow, Poland, November 2005, ESI preprint1756.

[3] L. Zajıcek, “Porosity and σ-porosity,” Real Analysis Exchange, vol. 13, no. 2, pp. 314–350, 1987-1988.[4] L. Zajıcek, “On σ-porous sets in abstract spaces,” Abstract and Applied Analysis, vol. 2005, no. 5, pp.

509–534, 2005, Proceedings of the International Workshop on Small Sets in Analysis, E. Matouskova,S. Reich and A. Zaslavski, Eds., Hindawi Publishing Corporation, New York, NY, USA.

[5] V. Olevskii, “A note on the Banach-Steinhaus theorem,” Real Analysis Exchange, vol. 17, no. 1, pp.399–401, 1991-1992.

[6] F. Strobin, “A comparison of two notions of porosity,” Commentationes Mathematicae, vol. 48, no. 2, pp.209–219, 2008.

[7] D. Preiss and L. Zajıcek, “Stronger estimates of smallness of sets of Frechet nondifferentiability ofconvex functions,” Rendiconti del Circolo Matematico di Palermo, Serie II, no. 3, supplement, pp. 219–223, 1984.

[8] W. Rudin, Functional Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill,New York, NY, USA, 2nd edition, 1991.

[9] J. Duda, “On the size of the set of points where the metric projection exists,” Israel Journal ofMathematics, vol. 140, pp. 271–283, 2004.

[10] P. Habala, P. Hajek, and V. Zizler, An Introduction to Banach Space Theory, vol. 1, MatFiz Press,University of Karlovy, Prague, Czech Republic, 1996.

[11] R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, vol. 1364 of Lecture Notes inMathematics, Springer, Berlin, Germany, 1989.

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