A SMALL COMPLEMENT

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A SMALL COMPLEMENT

JAROSLAV TIˇSER Received 29 October 2003

We show that in every Banach space, there is ag-porous set, the complement of which is ofᏴ¹-measure zero on everyC¹curve.

1. Introduction

The notion of porosity is one of the notions of smallness which had appeared in a natural way in the theory of diﬀerentiation of functions. Porous andσ-porous sets play the role of the exceptional sets in many contexts. A survey as well as further references on this topic can be found in [4]. In finite-dimensional Banach spaces, the family of allσ-porous sets is a proper subfamily of the sets of both 1st category and Lebesgue-measure zero.

The question how big a porous or aσ-porous set can be in an infinite-dimensional Banach space is closely related to the existence of a point of Fréchet differentiability of a Lipschitz function. The simplest connection may be the easy observation that for a porous setM, the distance functionf(x)=dist(x,M) is a Lipschitz function which is not Fréchet differentiable at any point ofM. Another more important link is represented by a deep result in [1]. From this result it follows, in particular, thatevery Lipschitz function on a separable Asplund space is Fréchet differentiable except for the points belonging to the union of aσ-porous set and aΓ-null set. The concept of aΓ-null set is another and relatively new concept of smallness of a set. We will not give here the definition (it can be found in the just-mentioned paper [1]) since it will not be used here. We only remark thatΓ-null sets form aσ-ideal of subsets and in a finite-dimensional space, theΓ-null sets are precisely the Lebesgue null sets.

There are spaces in whichσ-porous sets areΓ-null. Among the classical Banach spaces, the spacec0is such an example, see [1]. In those cases, the above-quoted statement gives a nice quantitative result saying that a Lipschitz function is Fréchet differentiable except for aΓ-null set. Unfortunately, the spacesp, 1< p <∞, do not belong to this class. The reason for that is an example published in [3]. It shows, though in an implicit way and with the help of the mean-value theorem from [1], that every spacep, 1< p <∞, neces- sarily contains aσ-porous set which is notΓ-null. Nevertheless, one can still profit from the result about the points of differentiability unless the correspondingσ-porous set is so

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big that its complement isΓ-null. A recent example in [2], however, makes clear that such a hugeσ-porous set does exist. Namely,there is aσ-porous set in a separable Hilbert space such that its complement intersects every three dimensional non degenerateC¹surface in a set of measure zero. This property is suﬃcient for a set to have aΓ-null complement. It is interesting to note that if we use two-dimensional surfaces instead of three-dimensional, the statement is false. For another result in [2], it was shown that given a porous setMin a separable Hilbert space, almost all points of a typical two-dimensional surface do not belong toM.

One of the most restrictive notions of smallness is the so-called unrectifiability. A set is called 1-purely unrectifiable if the intersection with anyC¹curve is ofᏴ¹-measure zero.

The space1contains aσ-porous set whose complement is 1-purely unrectifiable, see [2].

From what was said above, it is clear that such a bigσ-porous set cannot occur in every (separable) Banach space, for example not inc0. However, if we relax a bit the requirement for porosity, then it is possible to construct in every Banach space the “generalized”σ- porous set with 1-purely unrectifiable complement, seeTheorem 2.2below. An analogous result for standardσ-porous set holds true provided that we use only lines instead of all C¹curves, see [3]:in every separable Banach space, there is aσ-porous set the complement of which is of measure zero on every line.

Although the generalized porous set does not seem to have a direct connection to dif- ferentiation of Lipschitz functions, we believe that the result presented in the next section may be of some interest. At least it answers the question which naturally appeared in this context: how big can ag-porous set be?

2. Main result

We start with the notion of ag-porous set. This concept is a direct generalization of the standard porosity. It can be found also in the above-mentioned survey paper [4] together with the more detailed treatment of various other notions of porous sets. In what follows, we denote byB(x,r) the ball with centrexand radiusr.

Definition 2.1. Letg: [0,∞)→[0,∞) be a continuous increasing function, and letg(0)= 0. A setM⊂Xin a metric spaceXis calledg-porous if for anyx∈M,

lim sup

s→0+

gγ(x,s,M)

s >0, (2.1)

whereγ(x,s,M)=sup{r≥0|B(y,r)∩M= ∅, dist(x,y)≤s}.

A set which is a countable union ofg-porous sets is called aσ-g-porous set.

Everyg-porous set is a nowhere dense set and, consequently,σ-g-porous set is a set of the 1st category. Ifg(t)=t, theng-porosity becomes the standard porosity. Notice also that ifg₊(0) is finite and nonzero (or, more generally, 0< D₊g(0)≤D+g(0)<∞), then a g-porous set is again a porous set only. If we allowg₊(0)=+∞, then we get a type of sets which are in general bigger than porous sets. The next theorem deals exactly with such types ofg-porous sets.

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The symbolsᏴ¹andᏸ¹denote one-dimensional Hausdorﬀand Lebesgue measures, respectively.

Theorem2.2. LetXbe a Banach space and letg: [0,∞)→[0,∞)be a continuous, increas- ing function withg(0)=0and

lim sup

s→0+

g(s)

s ^{= ∞}. (2.2)

Then there is aσ-g-porous setM⊂Xsuch that the complementX\Mmeets everyC¹curve in a set of Ᏼ¹-measure zero. In short,X\Mis purely1-unrectifiable.

Remark 2.3. It is useful to realize the following equivalent reformulation of the conclusion ofTheorem 2.2. Ifϕ:R→Xis anyC¹curve, then

Ᏼ¹ϕ(R)∩(X\M)=0 (2.3)

if and only if

ᏸ¹t∈R|ϕ(t)∈/ M,ϕ(t)=0=0. (2.4) This is an immediate consequence of the formula for the length of a curve as follows:

Ᏼ¹ϕ(R)∩(X\M)=

{t|ϕ(t)∈/M}

ϕ(t)dᏸ¹(t)

=

{t|ϕ(t)∈/M,ϕ(t)=0_}

ϕ(t)dᏸ¹(t).

(2.5)

We prove first a simple lemma.

Lemma2.4. Let ϕ:R→X and let v∈X,v=0, be such thatϕ(t)−vtis a K-Lipschitz mapping with0≤K <v.

(i)For anys,t∈R,

v −K|s−t| ≤ϕ(s)−ϕ(t)≤

v+K|s−t|. (2.6) (ii)IfP⊂Xis a nonempty set, then

Ᏼ¹ϕ(R)∩P≤v+K

v −KdiamP. (2.7)

Remark 2.5. The assumption thatϕ(t)−vtis aK-Lipschitz mapping is a weakening of a more geometrically apparent condition which one can use in the case ofC¹mapping, namely,ϕ(t)−v ≤K. Indeed, let s,t∈R,s≤t, be arbitrary. Then the mean-value theorem gives

ϕ(s)−vs−

ϕ(t)−vt≤supϕ(θ)−v|θ∈(s,t)|s−t|

≤K|s−t|. (2.8)

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Proof. (i) Sinceϕ(t)−vtisK-Lipschitz, we have

ϕ(s)−ϕ(t)≤ϕ(s)−vs−ϕ(t)−vt+vs−vt

≤K|s−t|+v|s−t| =

v+K|s−t|. (2.9) Similarly, we estimate the expressionϕ(s)−ϕ(t)from below, which gives (i).

(ii) If diamP= ∞, the statement is clear. Assume diamP <∞. We denote t0=inft_∈_R_|ϕ(t)_∈P,

t1=supt∈R|ϕ(t)∈P. (2.10)

From already proved part (i), we infer that botht0andt1are finite and that v −Kt1−t0≤ϕt1

−ϕt0≤diamP. (2.11) Hence, using also the fact thatᏴ¹(ϕ([t0,t1]))≤(v+K)|t1−t0|, we obtain

Ᏼ¹ϕ(R)∩P=Ᏼ¹ϕt0,t1 ∩P

≤

v+Kt1−t0

≤v+K v₋KdiamP.

(2.12) Proof of Theorem 2.2. Consider the familyᏰof all 1-discrete sets inX. (We recall that a setDis calledλ-discrete ifd1−d2 ≥λwheneverd1,d2are two distinct elements in D.) By Zorn’s lemma,Ᏸcontains a maximal elementD. Further, the assumption (2.2) implies that there is a sequence (s_n)⊂R,s_n0, such that

rn≡ sn

gsn

0, ∞ n=1

rn<∞. (2.13)

Denote

Ꮾn=

Bd,r_n|d∈D, M_n=X\

k≥n

gs_kᏮk. (2.14) We show thatMnisg-porous. Letx∈Mn, and letk≥n. By the maximality ofD, there is d∈Dsuch that

x

gsk

−d<1, i.e., x−gsk

d< gsk

. (2.15)

The ballB(g(s_k)d,s_k)=g(s_k)B(d,r_k)⊂g(s_k)Ꮾk is obviously contained in the complement ofMn. Hence

γx,gsk

,Mn

≥sk. (2.16)

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The ratio of theg-porosity can be now estimated from below:

lim sup

s→0+

gγx,s,Mn

s ^≥lim sup

k→∞

gγx,gsk ,Mn gs_k

≥lim sup

k→∞

gs_k gs_k⁼1,

(2.17)

and theg-porosity ofMnis proved.

We putM=

nM_n. Clearly,Mis aσ-g-porous set.

Letϕ:R→Xbe anyC¹curve. In view ofRemark 2.3, we finish the proof by showing that the set

t∈R|ϕ(t)∈/ M,ϕ(t)=0 (2.18)

hasᏸ¹-measure zero. To this end, lett0 be an arbitrary point from this set and letv= ϕ(t0). Thenvis a nonzero vector. By continuity ofϕ, there is a compact intervalIcon- tainingt0in the interior and a number 0≤K <vverifying

ϕ(t)−v< K, t∈I. (2.19)

Since

ϕ(I)∩(X\M)=ϕ(I)∩ ∞ n=1

X\Mn

=ϕ(I)∩ ∞ n=1

k≥n

gsk

Ꮾk

⊂ϕ(I)∩

k≥n

gs_kᏮk

=

k≥n

ϕ(I)∩gsk Ꮾk

(2.20)

for anyn∈N, it follows that

Ᏼ¹ϕ(I)∩(X\M)≤

k≥n

Ᏼ¹ϕ(I)∩gsk

Ꮾk

. (2.21)

ByRemark 2.5, the mappingϕ(t)−vt isK-Lipschitz onI. Thus we can apply Lemma 2.4(ii) to each ballB(g(s_k)d,s_k) which meets the arcϕ(I) and we get

Ᏼ¹ϕ(I)∩Bgsk

d,sk

≤ v+K

v −K2sk. (2.22)

Now letn∈Nbe large enough to guarantee that 1≥g(sk)≥3sk for allk≥n. (This is possible due to (2.13).) Letk≥nand letN_kdenote the number of ballsB(g(s_k)d,s_k),d∈ D, intersecting the arcϕ(I). SinceDis 1-discrete, the distance between any two distinct

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points fromg(s_k)Dis at leastg(s_k). Hence the part of the arcϕ(I) between two consecutive balls has measure at leastg(s_k)−2s_k. So we infer that

N_k−1gs_k−2s_k≤Ᏼ¹ϕ(I)\gs_kᏮk

. (2.23)

The mappingϕitself is (v+K)-Lipschitz onIbyLemma 2.4(i). So Ᏼ¹ϕ(I)_\gs_kᏮk

≤Ᏼ¹ϕ(I)_≤v+Kᏸ¹(I). (2.24)

Combining (2.23) and (2.24), we obtain an upper estimate for the numberNk of balls meetingϕ(I):

Nk≤1 +

v+Kᏸ¹(I) gsk

−2sk . (2.25)

With the help of (2.22), we get Ᏼ¹ϕ(I)∩gsk

Ꮾk

=

d∈D

Ᏼ¹ϕ(I)∩Bgsk

d,sk

≤Nkv+K v −K2sk

≤

1 +

v+Kᏸ¹(I) gs_k−2s_k

v+K v −K2s_k

≤

1 +

v+Kᏸ¹(I) 2/3

v+K v −K

2s_k

gsk≤Crk,

(2.26)

where the constantCdepends only onᏸ¹(I) andK. Consequently, in view of (2.21), Ᏼ¹ϕ(I)∩(X\M)≤C

k≥n

rk. (2.27)

This is true for all n suﬃciently big. So we conclude that the intervalI possesses the property

Ᏼ¹ϕ(I)∩(X\M)=0, i.e., ᏸ¹t∈I|ϕ(t)∈/ M,ϕ(t)=0=0. (2.28) By separability ofR, the set{t∈R|ϕ(t)∈/ M,ϕ(t)=0}can be covered by countably many of such intervalsI. This fact, together with (2.28), finishes the proof.

Acknowledgment

The author was supported by Grants GA ˇCR 201/04/0090 and MSM 210000010.

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References

[1] J. Lindenstrauss and D. Preiss,On Fr´echet diﬀerentiability of Lipschitz maps between Banach spaces, Ann. of Math. (2)157(2003), no. 1, 257–288.

[2] J. Lindenstrauss, D. Preiss, and J. Tiˇser,Avoidingσ-porous sets, in preparation.

[3] D. Preiss and J. Tiˇser,Two unexpected examples concerning diﬀerentiability of Lipschitz functions on Banach spaces, Geometric Aspects of Functional Analysis (Israel, 1992–1994), Oper. The- ory Adv. Appl., vol. 77, Birkh¨auser, Basel, 1995, pp. 219–238.

[4] L. Zaj´ıˇcek,Porosity andσ-porosity, Real Anal. Exchange13(1987/1988), no. 2, 314–350.

Jaroslav Tiˇser: Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Zikova 4, 166 27 Praha 6, Czech Republic

E-mail address:[email protected]