# Porosity of Convex Nowhere Dense Subsets of Normed Linear Spaces

Authors: Filip Strobin
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

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


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## Body

## 1. Introduction

The paper concerns the topic of describing smallness of interesting sets of metric spaces in terms of porosity. The notions of porosity andσ-porosity (a set is σ-porous if it is a countable union of porous sets) can be considered as stronger versions of nowhere density and meagerness—in particular, in any “reasonable” metric space, there exist sets which are nowhere dense and are not σ-porous. Thus it is interesting to know that some sets are not only nowhere dense (meager) but even porous (σ-porous). In such a direction many earlier results were extended, for example, it turned out that the set of all Banach contractions was not only meager but also σ-porous in the space of all nonexpansive mappings (cf. [1, 2]). Since there are various types of porosity (more or less restrictive), the natural problem of finding the most restrictive notion of porosity, which would be suitable for an examined set, is also an interesting 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 approximation of smallness (in terms of porosity) of convex nowhere dense subsets of normed linear spaces? Zajíček [4] observed that such sets are R-ball porous for every R>0, and 0-cone porous (cf. [4, page 518]). In fact, Zajíček's observation is an improvement of the earlier result of Olevskii [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 would imply R-ball porosity for every R>0, and 0-cone porosity.The paper is organized as follows. In Section2, we give definitions of some types of porosity, that is, R-ball porosity, 0-angle porosity (a stronger version of 0-cone porosity) and introduce 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 characterization of smallness of convex nowhere dense sets.In Section3, we prove that in any Hilbert space H with dimH>1, the unit sphere is 0-angle porous and is not a countable union of c-porous sets (i.e., is not σ-c-porous). This observation shows that the notion of 0-angle porosity is quite far from the notion of c-porosity.The motivation for Section4 originates from the fact that the notion of c-porosity uses the space of all continuous functionals X*. In this section we discuss the possibility of finding the best approximation of smallness of a convex nowhere dense sets in terms of porosity without usingX*.In Section5, we give one example of σ-c-porous subset of the space of continuous functions. For other interesting σ-c-porous sets, we refer the reader to [5] (one of them deals with the Banach-Steinhaus principle).


## 2. Some Notions of Porosity

In this section we present the definitions ofR-ball porosity, ball smallness, (σ-)0-angle porosity, and (σ-)c-porosity. We also make some basic observations, which will be used in the 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, we denote by B(x,r) the open ball with center x and radius r. By X* we denote the space of all continuous linear functionals on X.Definition 2.1 (see [4, 7]).
LetR>0. We say that M is R-ball porous if for any x∈M and α∈(0,1), there exists y∈X such that ∥x-y∥=R andB(y,αR)∩M=∅.Remark 2.2.
The definition ofR-ball porosity presented in [7], [4, page 516] is slightly different from the above one. Namely, M is R-ball porous if for any x∈M and ε∈(0,R), there exists y∈X such that ∥x-y∥=R and B(y,R-ε)∩M=∅. However, it is obvious that both definitions are equivalent.Definition 2.3 (see [4, 7]).
We say thatM is 0-angle porous if for every x∈M and everyr>0, there exist y∈B(x,r) and ϕ∈X*∖{0}such that
(2.1){z∈X:ϕ(z)>ϕ(y)}∩M=∅.Note that0-angle porosity can be considered as a “global” version of (mentioned in the introduction) 0-cone porosity and, in particular, 0-angle porosity implies 0-cone porosity.For the definitions ofα-cone porosity and α-angle porosity, where α∈[0,1), see [4, page 516] and [7], respectively.Definition 2.4.
M is called c-porous if for any x∈X and every r>0, there are y∈B(x,r) and ϕ∈X*∖{0} such that
(2.2){z∈X:ϕ(z)>ϕ(y)}∩M=∅.C-porosity turns out to be the suitable notion to describe the smallness of convex nowhere dense sets (see Proposition2.5) and is a stronger form of 0-angle porosity (x∈X instead x∈M). Indeed, consider the unit sphere S of any nontrivial normed space. S is not c-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 setM is a countable union of c-porous sets, then we say that M is σ-c-porous. In the same way we define σ-0-angle porosity. If M=⋃n∈ℕMn and each Mn is Rn-ball porous for 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 the sense of porosity) of convex nowhere dense sets (in the proof we extend an argument suggested by Zajíček [4, page 518]).Proposition 2.5.
A subsetM of a normed space X is c-porous if and only if convM is nowhere dense.Proof.
“⇒” It is obvious that for any ϕ∈X* and y∈X, we have
(2.3){z:ϕ(z)>ϕ(y)}∩M=∅⇔{z:ϕ(z)>ϕ(y)}∩convM=∅.
Hence if M 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 any z∈convM¯,ϕ(z)<c. Then M∩{z:ϕ(z)>ϕ(y)}=∅.Corollary 2.6.
LetX be any normed space and let (*) be any condition such that
(2.4)ifA⊂Xsatisfies(*)andB⊂A,thenBsatisfies(*).
If every convex and nowhere dense subset of X satisfies (*), then any c-porous subset of X satisfies (*).The notions of0-angle porosity and c-porosity involve the space X*; however, in its origin the porosity was defined in metric spaces. In the next part of this section we will show what kind of porosity without using X* is implied by them (see Proposition 2.8). Note that we 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 deduced from the proof of [5, Lemma 1].Lemma 2.7.
LetR>0,  r∈(0,1/2),x0,y0∈X, ϕ∈X*∖{0}. If ∥y0-x0∥<rR, then there exists y∈X such that ∥y-x0∥=R and
(2.5)B(y,(1-2r)R)⊂{x∈X:ϕ(x)>ϕ(y0)}.Proposition 2.8.
The following statements hold.(i)
IfM is 0-angle porous, then M is R-ball porous for every R>0, that is,
(2.6)foreveryR>0,x∈Mandeveryα∈(0,1),thereexistsy∈Xwith∥y-x∥=RandB(y,αR)∩M=∅.(ii)
IfM is c-porous, then
(2.7)foreveryR>0,x∈Xandeveryα∈(0,1),thereexistsy∈Xwith∥y-x∥=RandB(y,αR)∩M=∅.Proof.
We will prove only(i), since the proof of (ii) is very similar. Fix R>0,  x0∈M and α∈(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 and M∩{x:ϕ(x)>ϕ(y0)}=∅. By Lemma 2.7, we have that there exists y∈X such that ∥y-x0∥=R and B(y,(1-2r)R)⊂{x:ϕ(x)>ϕ(y0)}. Since B(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 space satisfies (2.6) and does not satisfy (2.7). In the sequel, we will extend this observation (see Theorem 3.2).The next example shows, in particular, that the converse of the Proposition2.8 is not true.Example 2.9.
Let(X,∥·∥) be one of the following real Banach spaces: c0orlp,p∈[1,∞). Let us define the set M:=⋃n∈ℕ{±ne1,±ne2,…,±nen}, where
(2.8)en(r):={1,r=n,0,r≠n.Now we will show thatMsatisfies the following condition, which is stronger than (2.7) (and, in particular, than (2.6)):(2.9)foreveryR>0,x∈X,thereexistsy∈Xs.t.∥y-x∥=RandB(y,R)∩M=∅.To see it, take anyx∈X andR>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(2.10)y(k):={x(k),k≠n0,x(k)+R,k=n0.
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 thatM is not 0-angle porous. It is sufficient to show that for any ϕ∈X*∖{0} and y∈X, M∩{x:ϕ(x)>ϕ(y)}≠∅. Fix any ϕ∈X*∖{0} and y∈X, then there 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 that n2x0(n1)>ϕ(y). Then n2en1∈M and(2.11)ϕ(n2en1)=n2x0(n1)>ϕ(y).
Thus M is not 0-angle porous, and hence not c-porous.


## 3. Onc-Porosity

In this section we will show that c-porosity is a much stronger notion of porosity than0-angle porosity. This will justify introducing this notion.From now on, if(H,(·∣·)0) is a real Hilbert space, then (ℝ×H,(·∣·)) denotes the real Hilbert space ℝ×H with the inner product (·∣·) defined as follows:(3.1)((a,x)∣(b,y))≔ab+(x∣y)0forany(a,x),(b,y)∈ℝ×H.
Denote by ∥·∥0 and ∥·∥ the norms generated by (·∣·)0 and (·∣·), respectively.We will show that in any nontrivial real Hilbert spaceH 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 not σ-c-porous.Lemma 3.1.
Let(H,(·∣·)0) be a nontrivial real Hilbert space. For any δ∈(0,1), the set
(3.2)Vδ:={(a,y)∈ℝ×H:1-δ≤a≤1and∥(a,y)∥=1}
does not satisfy (2.7).Proof.
Take(1-(1/2)δ,0)∈ℝ×H, R=3 and
(3.3)α:=1Rmax{R2-4(2δ-δ2-12δ+116δ2),R2-14δ2}.
It is easy to see that α∈(0,1). Let (a0,y0)∈ℝ×H be such that
(3.4)∥(a0,y0)-(1-12δ,0)∥=R.
We will show that B((a0,y0),αR)∩Vδ≠∅. Consider the following three cases.Case 1 (a0≤1 and y0≠0).
Then(3.5)a0<-1or∥y0∥0>2.
Indeed, otherwise we would have a contradiction since
(3.6)R=∥(a0,y0)-(1-12δ,0)∥=(a0-(1-12δ))2+∥y0∥02=a02-2a0(1-12δ)+(1-12δ)2+∥y0∥02≤1+2+1+4<3=R.
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
(3.7)∥(1-δ,ηy0)-(a0,y0)∥2=(-12δ-(a0-(1-12δ)))2+∥(η-1)y0∥02=14δ2+δ(a0-(1-12δ))+(a0-(1-12δ))2+2δ-δ2-22δ-δ2∥y0∥0+∥y0∥02=R2-14δ2+(1+a0)δ-22δ-δ2∥y0∥0,
so if a0<-1, then, by (3.3), we infer
(3.8)∥(1-δ,ηy0)-(a0,y0)∥2<R2-14δ2≤α2R2,
and if ∥y0∥0>2, then, again by (3.3), we get
(3.9)∥(1-δ,ηy0)-(a0,y0)∥2<R2-14δ2+2δ-42δ-δ2≤α2R2.Case 2 (a0≤1 and y0=0).
In this casea0=(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
(3.10)∥(1-δ,z)-(a0,0)∥2=(1-δ-1+12δ+R)2+2δ-δ2=R2-δR+14δ2+2δ-δ2=R2-δ-34δ2<R2-34δ2<α2R2.Case 3 (a0>1).
Take(1,0)∈Vδ. By (3.3) and (3.4) we infer
(3.11)∥(a0,y0)-(1,0)∥2=∥((a0-(1-12δ))-12δ,y0)∥2=(a0-(1-12δ))2-δ(a0-(1-12δ))+14δ2+∥y0∥02=R2-δ(a0-1)-14δ2<R2-14δ2≤α2R2,
so (1,0)∈Vδ∩B((1-(1/2)δ,0),αR).
As a consequence, in all cases we haveB((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 in H. If S=⋃n∈ℕSn, then there is n0∈ℕ such that Sn0 does not satisfy (2.7). In particular, S is not σ-c-porous.Proof.
The second statement follows from the first one by Proposition2.8. We will prove the first statement. Let (H,(·∣·)0) be a Hilbert space with dimH>1. Since S is complete, by the Baire 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¯S we 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∈U and consider one-dimensional subspace M:={ax0:a∈ℝ}. It is well known (see, e.g., [8]) that
(3.12)M⊥:={y∈H:(y∣x0)0=0}
is a closed subspace of H and H=M⊕M⊥. Consider the space ℝ×M⊥. It is easy to see that the function H∋ax0+y↦F(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
(3.13)V:=F(U)={(a,y)∈ℝ×M⊥:ax0+y∈U}
does not satisfy (2.7) in ℝ×M⊥. Since S1:=F(S)={(a,y):∥(a,y)∥=1} andF∣S:S→S1 is a homeomorphism between S and S1, the set V is open in S1. Hence and by the fact that the point (1,0) is in V, we infer there exists 0<δ<1, such that
(3.14)Vδ⊂V,
where Vδ is defined as in Lemma 3.1. Indeed, since (1,0)∈V and V is open in S1, we have that there are c,d∈ℝ and r>0 such that 0<c<1<d and
(3.15)[(c,d)×B(0,r)]∩S1⊂V.
Set δ:=min{r2/2,(1-c)/2} and take any (a,y)∈Vδ, then
(3.16)d>1≥a≥1-δ>1-1+c=c,∥y∥0=1-a2≤1-(1-δ)2=2δ-δ2<2δ≤r,
so (a,y)∈[(c,d)×B(0,r)]∩S1 which yields (3.14). Since Vδ does not satisfy (2.7) in view of Lemma 3.1, the proof is completed.Now we show that for the Euclidean spaceℝ, all presented notions of porosity coincide. In [7, page 222] it is given that any ball small subset of ℝ is countable. Thus and by Proposition 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 of Porosity without UsingX*

In this section we will discuss the problem of finding the best approximation of smallness of a convex nowhere dense subset of a normed spaceX in terms of porosity without using X* (as was mentioned, in its origin porosity was defined as a strictly metric condition). By Propositions 2.5 and 2.8, any such set satisfies (2.7). This is a stronger version of the first part of Zajíček'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 a countable union of sets satisfying (2.7).Now letM be the set defined in Example 2.9. M satisfies (2.9), hence (2.7), and is not 0-angle porous, so is not c-porous. This shows that, in general, the notion of c-porosity is more 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), and hence, 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 for every R>0. The other are condition (2.9) and the following weakening of (2.9):(4.1)foranyR>0andanyx∈M,thereexistsy∈Xsuchthat(∥x-y∥=RandB(y,R)∩M=∅).Now the question arises whether any convex nowhere dense subset of any normed linear 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, Remark 2.4]),  and hence any convex nowhere dense subset of such space satisfies (2.9). However, in the remainder of this section we will show that in a very wide class of Banach spaces there are sets, 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 subspace M⊊X, which is not a countable union of sets satisfying (4.1).Proof.
SinceX is a nonreflexive Banach space, there exists a closed subspace M⊊X such that for every x0∈M,R>0 and every y∈X, if ∥y-x0∥=R, then B(y,R)∩M≠∅ (this is a well known fact which follows from the James' theorem [10, page 52]). We will show that M is not a countable union of sets satisfying (4.1). Assume that M=⋃n∈ℕMn. Since M is complete, by the Baire Category theorem, there exists n0∈ℕ such that Mn0 is not nowhere dense in M. Hence for some x∈M and r>0, we have that
(4.2)B(x,r)∩M⊂Mn0¯M=Mn0¯.
Since x∈Mn0¯, there exist x0∈Mn0 and r1>0 such that B(x0,r1)⊂B(x,r). Then
(4.3)B(x0,r1)∩M⊂Mn0¯.
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 hence B(y,R)∩Mn0≠∅.A natural question arises, what happens in reflexive spaces?Example 4.2.
An anonymous referee observed that the Hilbert cube(4.4)K={x∈l2:-1n≤x(n)≤1n,n∈ℕ}
does not satisfy (4.1). To see it, recall the concept of the so-called supported points. We say x∈M⊂X is a supported point of M, 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 point of K. Now assume that y∈l2 is such that ∥y-x0∥=∥y∥=1 and B(y,1)∩K=∅. Then by the Hahn-Banach separation theorem (cf. [8, page 38]), there exists ϕ∈l2*∖{0} with
(4.5)sup{ϕ(z):z∈K}≤inf{ϕ(z):z∈B(y,1)}.
On the other hand, x0 is on the boundary of B(y,1), and hence
(4.6)ϕ(x0)≥inf{ϕ(z):z∈B(y,1)}.
This gives a contradiction. Hence K does not satisfy (4.1).By Proposition4.1 and the previous example, condition (2.7) seems to be quite suitable for describing smallness of convex nowhere dense sets in terms of porosity without using X*. 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.
LetX=ℝ2,∥(x,y)∥=∣x∣+|y| and let
(4.7)M={(x,y):x≥12andx2+y2=1}.
It is easy to see that M satisfies (4.1). Moreover,using an analogous method as in the proof of Lemma 2.7, it can be easily shown that M is not σ-c-porous.


## 5. Applications

We will give one example ofσ-c-porous set (for other, we refer the reader to [5]).LetH be a Hilbert space, and let K 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}. Consider CB(K) as a Banach space with the norm ∥A∥:=supx∈K∥A(x)∥. Let 𝕜B be the set of all Banach contractions:(5.1)𝕜B={A:K→K:∃α∈(0,1)∀x,y∈K∥Ax-Ay∥≤α∥x-y∥}.
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
(5.2)Ω:={A:K→K:∀x,y∈K∥Ax-Ay∥≤∥x-y∥},
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-porous subset of CB(K). Since 𝕜B=⋃n∈ℕ𝕜B(1-1/n), the proof is completed.



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