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Porosity of Convex Nowhere Dense Subsets of Normed Linear Spaces
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.
The paper concerns the topic of describing smallness of interesting sets of metric spaces in terms of porosity. The notions of porosity and -porosity (a set is -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 -porous. Thus it is interesting to know that some sets are not only nowhere dense (meager) but even porous (-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 -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? Zajíek  observed that such sets are -ball porous for every and -cone porous (cf. [4, page 518]). In fact, Zajíek's observation is an improvement of the earlier result of Olevskii  (as was shown in , Olevskii worked with a much weaker version of porosity than -ball porosity). Hence for our purpose we need to find some stronger condition, which would imply -ball porosity for every and -cone porosity.
The paper is organized as follows. In Section 2, we give definitions of some types of porosity, that is, -ball porosity, -angle porosity (a stronger version of -cone porosity) and introduce the notion of c-porosity. We also make some basic observations (i.e., c-porosity -angle porosity -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 with the unit sphere is -angle porous and is not a countable union of c-porous sets (i.e., is not -c-porous). This observation shows that the notion of -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 . 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
In Section 5, we give one example of -c-porous subset of the space of continuous functions. For other interesting -c-porous sets, we refer the reader to  (one of them deals with the Banach-Steinhaus principle).
2. Some Notions of Porosity
In this section we present the definitions of -ball porosity, ball smallness, (-)-angle porosity, and (-)c-porosity. We also make some basic observations, which will be used in the sequel (see Proposition 2.8 and Example 2.9).
Let be a real normed linear space and . Given and , we denote by the open ball with center and radius . By we denote the space of all continuous linear functionals on .
Remark 2.2. The definition of -ball porosity presented in , [4, page 516] is slightly different from the above one. Namely, is -ball porous if for any and there exists such that and However, it is obvious that both definitions are equivalent.
Note that -angle porosity can be considered as a “global” version of (mentioned in the introduction) -cone porosity and, in particular, -angle porosity implies -cone porosity.
Definition 2.4. is called c-porous if for any and every there are and such that
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 -angle porosity ( instead ). Indeed, consider the unit sphere of any nontrivial normed space. is not c-porous (simply take and ) and is -angle porous—to see it, use the Hahn-Banach separation theorem (cf. ) for sets (the closure of ) and , where and .
If a set is a countable union of c-porous sets, then we say that is -c-porous. In the same way we define - -angle porosity. If and each is -ball porous for some , then we say that 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 Zajíek [4, page 518]).
Proposition 2.5. A subset of a normed space is c-porous if and only if conv is nowhere dense.
Proof. “” It is obvious that for any and we have
Hence if is c-porous, then conv is also c-porous and, in particular, nowhere dense.
“” Fix any and . Since (the closure of conv) is nowhere dense, there exists . Sets and satisfy the assumptions of the Hahn-Banach separation theorem, so there exist and such that and for any . Then .
Corollary 2.6. Let be any normed space and let be any condition such that If every convex and nowhere dense subset of satisfies , then any c-porous subset of satisfies .
The notions of -angle porosity and c-porosity involve the space ; 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 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 ].
Lemma 2.7. Let , . If then there exists such that and
Proposition 2.8. The following statements hold.(i)If is -angle porous, then is -ball porous for every , that is, (ii)If is c-porous, then
Proof. We will prove only since the proof of is very similar. Fix and . Let be such that , and let and be such that and By Lemma 2.7, we have that there exists such that and Since , the result follows.
The next example shows, in particular, that the converse of the Proposition 2.8 is not true.
Example 2.9. Let be one of the following real Banach spaces: or , . Let us define the set , where
To see it, take any and. Since , there exists such that . Assume, without loss of generality, that Now let be such that
Then . To see that , take any and consider three cases.(i), then (ii) then (iii), then
Now we will show that is not -angle porous. It is sufficient to show that for any and , . Fix any and , then there exists a sequence such that for any . Let be such that . Assume, without loss of generality, that . Let be such that . Then and
Thus is not -angle porous, and hence not c-porous.
3. On -Porosity
In this section we will show that c-porosity is a much stronger notion of porosity than -angle porosity. This will justify introducing this notion.
From now on, if is a real Hilbert space, then denotes the real Hilbert space with the inner product defined as follows:
Denote by and the norms generated by and respectively.
We will show that in any nontrivial real Hilbert space with , the unit sphere is not -c-porous. In fact, we will obtain a more general result. If , then there exists such that does not satisfy (2.7), and hence (by Proposition 2.8) is not -c-porous.
Lemma 3.1. Let be a nontrivial real Hilbert space. For any the set does not satisfy (2.7).
Proof. Take , and
It is easy to see that . Let be such that
We will show that Consider the following three cases.Case 1 ( and ). Then
Indeed, otherwise we would have a contradiction since
Set . It is easy to see that . We will show that . By (3.4), we have
so if then, by (3.3), we infer
and if then, again by (3.3), we get
Case 2 ( and ). In this case . Set with . It is obvious that . We will show that . By (3.3), (3.4), and a fact that , we get
Case 3 (). Take . By (3.3) and (3.4) we infer
As a consequence, in all cases we have and hence the result follows.
Theorem 3.2. Let be any Hilbert space with and let be the unit sphere in . If , then there is such that does not satisfy (2.7). In particular, is not -c-porous.
Proof. The second statement follows from the first one by Proposition 2.8. We will prove the first statement. Let be a Hilbert space with Since is complete, by the Baire Category theorem, there exists such that is not nowhere dense in Hence there exists a nonempty set open in such that (by we denote the closure of in the space ). Since the closure of a set which satisfies (2.7), also satisfies (2.7), the proof will be completed if we show that does not satisfy (2.7). Take any and consider one-dimensional subspace . It is well known (see, e.g., ) that is a closed subspace of and . Consider the space . It is easy to see that the function is an isometrical isomorphism between and . Since (2.7) is a metric condition, it suffices to show that the set does not satisfy (2.7) in . Since and is a homeomorphism between and , the set is open in . Hence and by the fact that the point is in we infer there exists such that where is defined as in Lemma 3.1. Indeed, since and is open in we have that there are and such that and Set and take any , then so which yields (3.14). Since does not satisfy (2.7) in view of Lemma 3.1, the proof is completed.
Now we show that for the Euclidean space , all presented notions of porosity coincide. In [7, page 222] it is given that any ball small subset of is countable. Thus and by Proposition 2.8, if , then is -c-porous is a countable union of sets satisfying (2.7) is --angle porous is ball small is countable.
4. Smallness of Convex Nowhere Dense Sets in Terms of Porosity without Using
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 in terms of porosity without using (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 Zajíek's observation, which states that such sets are -ball porous for every . Indeed, by Theorem 3.2, the unit sphere in Hilbert space is -ball porous for every and is not a countable union of sets satisfying (2.7).
Now let be the set defined in Example 2.9. satisfies (2.9), hence (2.7), and is not -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 -angle porous) does not satisfy (2.7), and hence, in general, the notion of -angle porosity and condition (2.7) are not comparable.
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 ]), 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 be a real nonreflexive Banach space. Then there exists a closed subspace which is not a countable union of sets satisfying (4.1).
Proof. Since is a nonreflexive Banach space, there exists a closed subspace such that for every and every , if , then (this is a well known fact which follows from the James' theorem [10, page 52]). We will show that is not a countable union of sets satisfying (4.1). Assume that . Since is complete, by the Baire Category theorem, there exists such that is not nowhere dense in . Hence for some and , we have that Since , there exist and such that . Then Fix any and let be such that . Then there exists , and then the segment . Hence if is such that , then . Thus , and hence .
A natural question arises, what happens in reflexive spaces?
Example 4.2. An anonymous referee observed that the Hilbert cube does not satisfy (4.1). To see it, recall the concept of the so-called supported points. We say is a supported point of , if there exists such that ; if such a functional does not exist, then is called a nonsupported point (cf. [11, page 44]). Now take and . Then it is easy to see that is a nonsupported point of . Now assume that is such that and . Then by the Hahn-Banach separation theorem (cf. [8, page 38]), there exists with On the other hand, is on the boundary of , and hence This gives a contradiction. Hence 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 . 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.
We will give one example of -c-porous set (for other, we refer the reader to ).
Let be a Hilbert space, and let be a nonempty bounded closed and convex subset of Define is continuous and is bounded in Consider as a Banach space with the norm . Let be the set of all Banach contractions:
For any we also define
Proposition 5.1. is a -c-porous subset of In particular, is ball small.
Proof. De Blasi and Myjak  (cf. also ) proved that for any , is lower porous (and hence nowhere dense; for the definition of lower porosity, see ) subset of the space with the metric induced from . Hence is a nowhere dense subset of It is also obvious that is convex. As a consequence, for any the set is a c-porous subset of . Since the proof is completed.
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 Gąb and Jacek Jachymski for many valuable discussions.
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Copyright © 2009 Filip Strobin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.