Porosity of Convex Nowhere Dense Subsets of Normed Linear Spaces

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.

1. Introduction

The paper concerns the topic of describing smallness of interesting sets of metric spaces in terms of porosity. The notions of porosity and 𝜎-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 Section 2, 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 Section 3, 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 Section 4 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 using 𝑋^∗.

In Section 5, 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 of 𝑅-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]). Let 𝑅>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 of 𝑅-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 that 𝑀 is 0-angle porous if for every 𝑥∈𝑀 and every𝑟>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 in the 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.

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 {𝑧∈𝑋∶𝜙(𝑧)>𝜙(𝑦)}∩𝑀=∅.(2.2)

C-porosity turns out to be the suitable notion to describe the smallness of convex nowhere dense sets (see Proposition 2.5) 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 set 𝑀 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∈ℕ𝑀_𝑛 and each 𝑀_𝑛 is 𝑅_𝑛-ball porous for some 𝑅_𝑛>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 subset 𝑀 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 {𝑧∶𝜙(𝑧)>𝜙(𝑦)}∩𝑀=∅⇔{𝑧∶𝜙(𝑧)>𝜙(𝑦)}∩conv𝑀=∅.(2.3) 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. Let 𝑋 be any normed space and let (∗) be any condition such that if𝐴⊂𝑋satisfies(∗)and𝐵⊂𝐴,then𝐵satisfies(∗).(2.4) If every convex and nowhere dense subset of 𝑋 satisfies (∗), then any c-porous subset of 𝑋 satisfies (∗).

The notions of 0-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. Let 𝑅>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 𝐵(𝑦,(1−2𝑟)𝑅)⊂{𝑥∈𝑋∶𝜙(𝑥)>𝜙(𝑦₀)}.(2.5)

Proposition 2.8. The following statements hold.(i)If 𝑀 is 0-angle porous, then 𝑀 is 𝑅-ball porous for every 𝑅>0, that is, forevery𝑅>0,𝑥∈𝑀andevery𝛼∈(0,1),thereexists𝑦∈𝑋with‖𝑦−𝑥‖=𝑅and𝐵(𝑦,𝛼𝑅)∩𝑀=∅.(2.6)(ii)If 𝑀 is c-porous, then forevery𝑅>0,𝑥∈𝑋andevery𝛼∈(0,1),thereexists𝑦∈𝑋with‖𝑦−𝑥‖=𝑅and𝐵(𝑦,𝛼𝑅)∩𝑀=∅.(2.7)

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 Proposition 2.8 is not true.

Example 2.9. Let (𝑋,‖⋅‖) be one of the following real Banach spaces: 𝑐₀ or 𝑙_𝑝, 𝑝∈[1,∞). Let us define the set 𝑀∶=⋃_𝑛∈ℕ{±𝑛𝑒₁,±𝑛𝑒₂,…,±𝑛𝑒_𝑛}, where 𝑒_𝑛(𝑟)∶=1,𝑟=𝑛,0,𝑟≠𝑛.(2.8)

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

forevery𝑅>0,𝑥∈𝑋,thereexists𝑦∈𝑋s.t.‖𝑦−𝑥‖=𝑅and𝐵(𝑦,𝑅)∩𝑀=∅.(2.9)

To see it, take any 𝑥∈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

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 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=1𝑥(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 𝑛₂𝑒_𝑛1∈𝑀 and

𝜙(𝑛₂𝑒_𝑛1)=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-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:

((𝑎,𝑥)∣(𝑏,𝑦))∶=𝑎𝑏+(𝑥∣𝑦)₀forany(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∈ℕ𝑆_𝑛, then there exists 𝑛₀∈ℕ such that 𝑆_𝑛0 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 𝑉_𝛿∶={(𝑎,𝑦)∈ℝ×𝐻∶1−𝛿≤𝑎≤1and‖(𝑎,𝑦)‖=1}(3.2) does not satisfy (2.7).

Proof. Take (1−(1/2)𝛿,0)∈ℝ×𝐻, 𝑅=3 and 1𝛼∶=𝑅max{𝑅²−4(2𝛿−𝛿²−121𝛿+𝛿16²),𝑅²−14𝛿²}.(3.3) It is easy to see that 𝛼∈(0,1). Let (𝑎₀,𝑦₀)∈ℝ×𝐻 be such that ‖(𝑎₀,𝑦₀1)−(1−2𝛿,0)‖=𝑅.(3.4) We will show that 𝐵((𝑎₀,𝑦₀),𝛼𝑅)∩𝑉_𝛿≠∅. Consider the following three cases.Case 1 (𝑎₀≤1 and 𝑦₀≠0). Then 𝑎₀<−1or‖𝑦₀‖₀>2.(3.5) Indeed, otherwise we would have a contradiction since 𝑅=‖(𝑎₀,𝑦₀1)−(1−2𝛿,0)‖=(𝑎₀1−(1−2𝛿))²+‖𝑦₀‖²₀=𝑎²₀−2𝑎₀1(1−21𝛿)+(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)∈𝑉_𝛿. We will show that (1−𝛿,𝜂𝑦₀)∈𝐵((𝑎₀,𝑦₀),𝛼𝑅). By (3.4), we have ‖(1−𝛿,𝜂𝑦₀)−(𝑎₀,𝑦₀)‖²1=(−2𝛿−(𝑎₀1−(1−2𝛿)))²+‖(𝜂−1)𝑦₀‖²₀=14𝛿²+𝛿(𝑎₀1−(1−2𝛿))+(𝑎₀1−(1−2𝛿))²+2𝛿−𝛿²−22𝛿−𝛿²‖𝑦₀‖₀+‖𝑦₀‖²₀=𝑅²−14𝛿²+(1+𝑎₀)𝛿−22𝛿−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<𝑅²−14𝛿²+2𝛿−42𝛿−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)∈𝑉_𝛿. 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)‖²1=(1−𝛿−1+2𝛿+𝑅)²+2𝛿−𝛿²=𝑅²1−𝛿𝑅+4𝛿²+2𝛿−𝛿²=𝑅²3−𝛿−4𝛿²<𝑅²−34𝛿²<𝛼²𝑅².(3.10)Case 3 (𝑎₀>1). Take (1,0)∈𝑉_𝛿. By (3.3) and (3.4) we infer ‖(𝑎₀,𝑦₀)−(1,0)‖²=‖((𝑎₀1−(1−21𝛿))−2𝛿,𝑦₀)‖²=(𝑎₀1−(1−2𝛿))²−𝛿(𝑎₀1−(1−21𝛿))+4𝛿²+‖𝑦₀‖²₀=𝑅²−𝛿(𝑎₀1−1)−4𝛿²<𝑅²−14𝛿²≤𝛼²𝑅²,(3.11) so (1,0)∈𝑉_𝛿∩𝐵((1−(1/2)𝛿,0),𝛼𝑅).
As a consequence, in all cases we have 𝐵((𝑎₀,𝑦₀),𝛼𝑅)∩𝑉_𝛿≠∅, 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∈ℕ𝑆_𝑛, then there is 𝑛₀∈ℕ such that 𝑆_𝑛0 does not satisfy (2.7). In particular, 𝑆 is not 𝜎-c-porous.

Proof. The second statement follows from the first one by Proposition 2.8. 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 𝑆_𝑛0 is not nowhere dense in 𝑆. Hence there exists a nonempty set 𝑈 open in 𝑆 such that 𝑈⊂𝑆_𝑛0^𝑆=𝑆_𝑛0 (by 𝑆_𝑛0^𝑆 we denote the closure of 𝑆_𝑛0 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 𝑀^⟂∶={𝑦∈𝐻∶(𝑦∣𝑥₀)₀=0}(3.12) 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 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} 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 𝑉_𝛿⊂𝑉,(3.14) where 𝑉_𝛿 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 [(𝑐,𝑑)×𝐵(0,𝑟)]∩𝑆₁⊂𝑉.(3.15) Set 𝛿∶=min{𝑟²/2,(1−𝑐)/2} and take any (𝑎,𝑦)∈𝑉_𝛿, 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 𝑉_𝛿 does not satisfy (2.7) in view of Lemma 3.1, the proof is completed.

Now we show that for the Euclidean space ℝ, 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 Using 𝐗^∗

In this section we will discuss the problem of finding the best approximation of smallness of a convex nowhere dense subset of a normed space 𝑋 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 let 𝑀 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):

forany𝑅>0andany𝑥∈𝑀,thereexists𝑦∈𝑋suchthat(‖𝑥−𝑦‖=𝑅and𝐵(𝑦,𝑅)∩𝑀=∅).(4.1)

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. Since 𝑋 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∈ℕ𝑀_𝑛. Since 𝑀 is complete, by the Baire Category theorem, there exists 𝑛₀∈ℕ such that 𝑀_𝑛0 is not nowhere dense in 𝑀. Hence for some 𝑥∈𝑀 and 𝑟>0, we have that 𝐵(𝑥,𝑟)∩𝑀⊂𝑀_𝑛0^𝑀=𝑀_𝑛0.(4.2) Since 𝑥∈𝑀_𝑛0, there exist 𝑥₀∈𝑀_𝑛0 and 𝑟₁>0 such that 𝐵(𝑥₀,𝑟₁)⊂𝐵(𝑥,𝑟). Then 𝐵(𝑥₀,𝑟₁)∩𝑀⊂𝑀_𝑛0.(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)∩𝑀_𝑛0. Thus 𝐵(𝑦,𝑅)∩𝑀_𝑛0≠∅, and hence 𝐵(𝑦,𝑅)∩𝑀_𝑛0≠∅.

A natural question arises, what happens in reflexive spaces?

Example 4.2. An anonymous referee observed that the Hilbert cube 𝐾={𝑥∈𝑙₂1∶−𝑛1≤𝑥(𝑛)≤𝑛,𝑛∈ℕ}(4.4) 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∈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 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. Let 𝑋=ℝ², ‖(𝑥,𝑦)‖=∣𝑥∣+|𝑦| and let 1𝑀={(𝑥,𝑦)∶𝑥≥√2and𝑥²+𝑦²=1}.(4.7) 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 of 𝜎-c-porous set (for other, we refer the reader to [5]).

Let 𝐻 be a Hilbert space, and let 𝐾 be a nonempty bounded closed and convex subset of 𝐻. Define 𝐶^𝐵(𝐾)∶={𝐴∶𝐾→𝐻∶𝐴 is continuous and 𝐴(𝐾) is bounded in 𝐻}. Consider 𝐶^𝐵(𝐾) as a Banach space with the norm ‖𝐴‖ ∶=sup_𝑥∈𝐾‖𝐴(𝑥)‖. Let 𝕜𝐵 be the set of all Banach contractions:

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 𝕜𝐵_𝛼∶={𝐴∶𝐾→𝐾∶∀_{𝑥,𝑦∈𝐾} ‖𝐴𝑥−𝐴𝑦‖≤𝛼‖𝑥−𝑦‖}.

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

Proof. De Blasi and Myjak [1] (cf. also [2]) proved that for any 𝛼<1, 𝕜𝐵_𝛼 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 (𝐶^𝐵(𝐾),‖⋅‖). Hence 𝕜𝐵_𝛼 is a nowhere dense subset of 𝐶^𝐵(𝐾). It is also obvious that 𝕜𝐵_𝛼 is convex. As a consequence, for any 𝛼<1, the set 𝕜𝐵_𝛼 is a c-porous subset of 𝐶^𝐵(𝐾). Since 𝕜𝐵=⋃_𝑛∈ℕ𝕜𝐵_(1−1/𝑛), the proof is completed.

Acknowledgments

The author is very grateful to the anonymous referees and Simeon Reich for suggestions how to modify the paper, and for solving the problem concerning condition (4.1) in reflexive spaces. Also, he would like to thank Szymon Głąb and Jacek Jachymski for many valuable discussions.