The set of space-filling curves: topological and algebraic structure

arXiv:1407.3951v1 [math.GN] 15 Jul 2014 THE SET OF SPACE-FILLING CURVES: TOPOLOGICAL AND ALGEBRAIC STRUCTURE L. BERNAL-GONZ´ ALEZ, M.C. CALDER ´ ON-MORENO AND J.A. PRADO-BASSAS Abstract. In this paper, a study of topological and algebraic properties of two families of functions from the unit interval Iinto the plane R2is performed. The first family is the collection of all Peano curves, that is, of those continuous mappings onto the unit square. The second one is the bigger set of all space-filling curves, i.e. of those continuous functions I→R2whose images have positive Jordan content. Emphasis is put on the size of these families, in both topological and algebraic senses, when endowed with natural structures. 1. Introduction In 1890 G. Peano [25] showed the existence of an astonishing mathematical object, namely, a curve filling the unit square. To be more precise, he constructed a continuous surjective mapping I→I2, where I= [0,1] is the closed unit interval in the real line Rand I2= [0,1] ×[0,1]. Lebesgue [15,16,22] was probably the first to show an example of a function f:R→Rthat is surjective in a strong sense. Specifically, it satisfies f(J) = Rfor every nondegenerate interval J. Since then, many families of surjections R→R, even in much stronger senses, have been presented (see [14, 19, 20]). Nevertheless, each of these functions is nowhere continuous. Of course, by using a bijection R→I2or R→R2, surjections R→I2or R→R2(or even I→I2) can be constructed, but their continuity is far from being guaranteed. Peano’s result admits a topological extension, and in fact a topological characterization, which is given by the Hahn–Mazurkiewicz theorem (see e.g. [30, Theorem 31.5] or [18]): a Hausdorff topological space Yis a continuous image of the unit interval if and only if it is a compact, connected, locally connected, and second-countable space. Such a space Yis called a Peano space. Equivalently, by well-known metrization theorems, a Peano space is a compact, connected, locally connected metrizable topological space. Given 2010 Mathematics Subject Classification. 15A03, 20M05, 46E15, 54E40, 54F15. Key words and phrases. Peano curve, space-filling curve, lineability, spaceability, algebrability. This paper is dedicated to Professor Jos´e Bonet Solves on his 60th birthday. The authors have been partially supported by the Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa FQM-127 Grant P08-FQM-03543 and by MEC Grant MTM201234847-C02-01. 1 2 BERNAL-GONZ ´ ALEZ, CALDER ´ ON-MORENO AND PRADO-BASSAS two topological spaces Xand Y, the set of continuous (continuous surjective, resp.) mappings X→Ywill be denoted by C(X, Y ) (CS(X, Y ), resp.). Then the family of Peano curves is P:= CS(I, I2). If Yis a Peano space, we also denote PY:= CS(I, Y ), so that P=PI2. There are several extensions of the notion of Peano curve on RN, with N≥2. Since the case N= 2 is illuminating enough, we will restrict ourselves to it. For instance, in [26], the next notion is given. By c(A) it is denoted the Jordan content of a Jordan measurable set A⊂R2(see Section 2 for definitions). Definition 1.1. We say that a continuous function ϕ:I→R2is a spacefilling curve provided that ϕ(I) is Jordan measurable and c(ϕ(I)) >0. We can relax this condition by defining a λ-space-filling curve –where λ denotes Lebesgue measure on R2– as a continuous function f:I→R2 with λ(f(I)) >0. This is not equivalent to the former definition; as a matter of fact, Osgood [24,26] constructed in 1903 a Jordan curve, that is, a continuous injective function ψ:I→R2, such that λ(ψ(I)) >0; here ψ(I) cannot be Jordan measurable. Other related notions can be found in [23] and [29]. The symbol SF will stand for the set of all space-filling curves in the sense of Definition 1.1. The main concern of this paper is to study both families Pand SF from the topological-algebraic point of view, with special emphasis on the size of such sets, rather than on properties of individual members of them. For this, Pand SF are supposed endowed with their natural topologies. The diverse notions of largeness that will be considered, together with other preliminaries, are compiled in Section 2. Finally, Sections 3 and 4 contain our main results, which demonstrate the existence of large –topological or algebraic– structures within the mentioned families. 2. Topological and linear size concepts When dealing with subsets of a metric space (X, d), one way to describe their smallness is by means of the notion of porosity, introduced by Dolzenko [13] in 1967 for the real line and generalized by Zaj´ıˇcek [31]. Here we use a slightly stronger notion of porosity [32]. By B(x, r) we denote the open ball in Xwith center x∈Xand radius r > 0, while Astands for the closure of a set Ain a given topological space. Definition 2.1. A subset Ain a metric space (X, d) is called porous if there is α > 0 such that for each x∈Xand each ε > 0 there exists y∈B(x, ε) such that B(y, α d(x, y)) ∩A=∅. If the above number α > 0 can be chosen as close to 1 as we wish then Ais called strongly porous. SPACE-FILLING CURVES: TOPOLOGICAL AND ALGEBRAIC STRUCTURE 3 It is well known that any porous set Ais nowhere dense, that is, its interior A0=∅. In fact, porosity is a notion strictly stronger than nowhere density. Porosity will be considered in the context of Peano curves. In a completely metrizable topological space X(so that Baire’s theorem applies), one way to describe smallness or largeness is by meagerness: a subset A⊂Xis said to be meager or of first category if it is a countable union of nowhere dense subsets; and Ais called residual if the complement X\Ais meager or, equivalently, if Ais a countable intersection of dense open sets. Hence, in a topological sense, a residual set is very large, and in fact the existence of many “strange” mathematical objects has been stated by proving that their set is residual (in some appropriate topological space). Incidentally, each set of such mathematical objects turns to be huge. A different, recently introduced approach to study the size of a family of objects arises from the theory of lineability. The following notions can be found in [4–7,9,11,17,27]. Definition 2.2. If Xis a vector space, αis a cardinal number and A⊂X, then Ais said to be: •lineable if there is an infinite dimensional vector space Msuch that M\ {0} ⊂ A, •α-lineable if there exists a vector space Mwith dim(M) = αand M\ {0} ⊂ A(hence lineability means ℵ0-lineability, where ℵ0= card (N), the cardinality of the set of positive integers), and •maximal lineable in Xif Ais dim (X)-lineable. If, in addition, Xis a topological vector space, then Ais said to be: •dense-lineable in Xwhenever there is a dense vector subspace Mof Xsatisfying M\ {0} ⊂ A, •maximal dense-lineable in Xwhenever there is a dense vector subspace Mof Xsatisfying M\ {0} ⊂ Aand dim (M) = dim (X), and •spaceable in Xif there is a closed infinite dimensional vector subspace Msuch that M\ {0} ⊂ A. When Xis a topological vector space contained in some (linear) algebra then Ais called: •algebrable if there is an algebra Mso that M\{0} ⊂ Aand Mis infinitely generated, that is, the cardinality of any system of generators of Mis infinite, and •strongly algebrable if, in addition, the algebra Mcan be taken free. Note that if Xis contained in a commutative algebra then a set B⊂X is a generating set of some free algebra contained in Aif and only if for any N∈N, any nonzero polynomial Pin Nvariables without constant term and any distinct f1, ..., fN∈B, we have P(f1, ..., fN)6= 0 and P(f1, ..., fN)∈A. 4 BERNAL-GONZ ´ ALEZ, CALDER ´ ON-MORENO AND PRADO-BASSAS Observe that strong-algebrability ⇒algebrability ⇒lineability, and none of these implications can be reversed, see [6] and [11, p. 74]. From Peano’s result, it is not difficult to extend his filling curve I→I2 to a continuous surjective function R→R2. This can be generalized as to obtain that CS(Rm,Rn)6=∅for all m, n ∈N. In fact, Albuquerque, Bernal, Ord´o˜nez, Pellegrino and Seoane [1, 2, 10] have recently shown that CS(Rm,Rn) is maximal dense-lineable and spaceable in C(Rm,Rn), and that CS(Rm,Cn) is strongly c-algebrable (here cstands for the cardinality of the continuum, Cdenotes the complex field, and the algebra structure of C(Rm,Cn) is defined coordenatewise). In [2], the lineability of the families CS(Rm, Y ), where Yrepresents some relevant subspaces of infinite dimensional Euclidean spaces, is also analyzed. To summarize, these diverse CS-families are large in several algebraic (or topological-algebraic) senses. It must be said that the mentioned results in [1,2,10] were the inspiration for the present paper, but there is an important point which is why the methods given in them cannot be directly reproduced in our setting. Namely, our starting space is the compact interval I. Hence f(I) is compact for any continuous mapping on I, so f(I) is never “too much large”. Furthermore, our family Pis not even stable under scaling, which causes that the study of lineability of Pmakes no sense. In order to investigate the algebraic structure of P, let us introduce the following concept. Definition 2.3. Assume that (X, ∗) is a semigroup and that A⊂X. We say that Ais semigroupable whenever there exists an infinitely generated semigroup G⊂A. Remark 2.4. We recall that a semigroup Gis called infinitely generated whenever it is not finitely generated, that is, there does not exist a finite set F⊂Xsuch that every x∈Gcan be written as a finite product x= am1 1∗ · · · ∗ amp p, with a1,...,ap∈Fand m1,...,mp∈N(of course, p,ai and midepend upon x). The ai’s are not necessarily different: take into account that (X, ∗) might be noncommutative. Nevertheless, the semigroup Xthat will be considered in this paper is C(I, I2), where the operation ∗ is the coordenatewise multiplication, which is commutative. Hence the ai’s can be taken different in this case. Recall that if Eis a Banach space then a sequence {xn}n≥1is called a basic sequence whenever it is a Schauder basis of its generated closed vector subspace, that is, whenever every vector x∈span{xn}n≥1can be uniquely represented by a series x=Pn≥1λnxnconverging in the norm k · k of E. By Nikolskii’s theorem (see for instance [12]), a sequence {xn}n≥1⊂E\{0} is basic if and only if there is a constant α∈(0,+∞) such that, for every pair r, s ∈Nwith s≥rand every finite sequence of scalars a1,...,as, one SPACE-FILLING CURVES: TOPOLOGICAL AND ALGEBRAIC STRUCTURE 5 has   r X n=1 anxn ≤α  s X n=1 anxn . For any N∈N, we will consider the norm kfk= supt∈Ikf(t)k1in the space C(I, RN), which makes it a Banach space; here k · k1represents the 1-norm in RN, given by k(x1,...,xN)k1= max1≤i≤N|xi|. In Section 4 the next lemma –which is a direct application of Nikolskii’s theorem– will be needed. Lemma 2.5. Assume that {fn}n≥1is a sequence in C(I, RN)\ {0}such that the supports {t∈I:fn(t)6= 0}(n= 1,2,...)are mutually disjoint. Then {fn}n≥1is a basic sequence in C(I, RN). The next assertion –which is proved in [10, Theorem 2.3] (see also [3,8,9])– will be useful in Section 4 to get dense-lineability from mere lineability. Theorem 2.6. Assume that Eis a metrizable separable topological vector space and that αis an infinite cardinal number. Let A, B ⊂Ebe two subsets such that Ais α-lineable, Bis dense-lineable, A∩B=∅and A+B⊂A. Then Acontains a dense vector space Mwith dim(M) = α. The following elementary lemma will be used repeatedly along Sections 3–4. Lemma 2.7. Let Ybe a Peano space and [a, b]be a closed interval in R. Given u, v ∈Y, there is a mapping Φ∈CS([a, b], Y )such that Φ(a) = u and Φ(b) = v. Proof. By the Hahn–Mazurkiewicz theorem, we can select a mapping f∈ PY. Since Peano spaces are arcwise connected [30, Theorem 31.2], there are continuous mappings g: [0,1/3] →Yand h: [2/3,1] →Ysatisfying g(0) = u,g(1/3) = f(0), h(2/3) = f(1) and h(1) = v. Define ϕ:I→Yas ϕ(t) =    g(t) if 0 ≤t < 1/3 f(3t−1) if 1/3≤t≤2/3 h(t) if 2/3≤t≤1. Then it is evident that the mapping Φ : [a, b]→Ygiven by Φ(t) = ϕt−a b−a does the job.  Finally, let us recall a number of concepts concerning the Jordan measurability. Assume that Sis a bounded subset of R2. Then the inner Jordan content and the outer Jordan content of Sare respectively given by the following lower and upper Riemann integrals: c(S) = ZχSdxdy, c(S) = ZχSdxdy, where χSdenotes the characteristic function of S. The set Sis said to be Jordan measurable provided that c(S) = c(S), in which case their common value c(S) is called the Jordan content of S. This happens if and only 6 BERNAL-GONZ ´ ALEZ, CALDER ´ ON-MORENO AND PRADO-BASSAS if χSis Riemann integrable, and if and only if λ(∂S) = 0 (∂S denotes the boundary of S). Moreover, in this case, Sis Lebesgue measurable and c(S) = λ(S). 3. The family of Peano curves A natural, complete distance on the space C(I, R2) is given by ρ(f, g) = sup t∈I d∞(f(t), g(t)),(1) that generates the topology of uniform convergence on I. Here d∞is the metric on R2resulting from the 1-norm k · k1, that is, d∞((a, b),(c, d)) = max{|a−c|,|b−d|} (other equivalent, even similar, metrics are available on R2, but d∞is more convenient for the sake of calculations). Of course, Pis a very small subset of C(I, R2). The main reason for it is that f(I) = I2for each f∈ P. This is why it is more natural to consider Pas a topological subspace of C(I, I2) rather than of C(I, R2). Observe that, due to the fact that uniform convergence entails pointwise convergence, C(I, I2) (endowed with the distance ρinduced from C(I, R2)) is closed in C(I, R2) (in fact, C(I, A) is closed in C(I, R2), for every closed set A⊂R2), so it is a complete metric space. For a general Peano space Y, it will be endowed with a fixed distance d generating its topology (note that, as Yis compact, any distance generating its topology is complete). Then, just by changing d∞to d, the expression (1) above defines a complete distance on C(I, Y ). Observe that, since Y is metrizable and arcwise connected, it is uncountable as soon as it possesses more than one point; in fact, every nonempty open subset of Yis uncountable. In the following theorem, we gather some topological or metrical properties of P. We use standard notation for a metric space (X, D): BD(x0, r) and BD(x0, r) will stand, respectively, for the open ball and the closed ball with center x0∈Xand radius r > 0. Theorem 3.1. Assume that Yis a Peano space. We have: (a) PYis closed in C(I, Y ). In particular, PYis a completely metrizable space. (b) If Yhas at least two points then PYis not compact. (c) Assume that Yhas at least two points and that there is y0∈Y satisfying the following property: given a neighborhood Uof y0, there exists a neighborhood Vof y0such that V⊂Uand V\ {y0}is arcwise connected. Then P0 Y=∅. Hence PYis nowhere dense in C(I, Y ). (d) In the case Y=I2, the Peano family PY=Pis strongly porous in C(I, I2). Proof. (a) Let F∈C(I, Y ) and {fn}n≥1be a sequence in PYwith fn→F. Fix y∈Y. Then there is a sequence {tn}n≥1⊂Isuch that fn(tn) = yfor all n∈N. Since Iis compact, we can take out a subsequence SPACE-FILLING CURVES: TOPOLOGICAL AND ALGEBRAIC STRUCTURE 7 {tnk}k≥1converging to some point t0∈I. The continuity of Fyields αk:= d(F(tnk), F(t0)) →0 as k→ ∞. From the triangle inequality, d(y, F (t0)) ≤d(fnk(tnk), F(tnk)) + d(F(tnk), F(t0)) ≤ρ(fnk, F) + αk−→ 0. Hence d(y, F(t0)) = 0 or, that is the same, F(t0) = y. Since ywas arbitrary, Fis surjective, that is, F∈ PY. Therefore PYis closed. (b) Choose y, z ∈Ywith y6=z. Set ε:= d(y, z)>0 and fix δ∈(0,1). By Lemma 2.7, there exists a continuous surjective function Φ : [0, δ/2] →Y with Φ(0) = y= Φ(δ/2). In particular, there is v∈[0, δ/2] such that Φ(v) = z. By extending Φ as yon (δ/2,1] and setting u:= 0, we obtain points u, v ∈Iand a mapping Φ ∈ PYsuch that |u−v|< δ but d(Φ(u),Φ(v)) ≥ ε. In other words, the family PYis not equicontinuous. According to the generalized Arzel´a theorem (see e.g. [21, pp. 119–120]), PYcannot be relatively compact, so it is not compact. (c) Consider the point y0given in the hypothesis and suppose, by way of contradiction, that P0 Y6=∅. Then there are f∈ PYand r > 0 such that Bρ(f, r)⊂ PY. In other words, if g∈C(I, Y ) and ρ(g, f)< r then g(I) = Y. On one hand, a neighborhood Vof y0can be found such that Bd(y0, r/2) ⊃Vand V\{y0}is arcwise connected. On the other hand, there is a closed ball Bd(y0, s)⊂V. Since fis continuous, the set f−1(Bd(y0, s)) is open in I, so it is a countable union of pairwise disjoint intervals of the form (α, β), [0, β) or (α, 1]. In all three cases, we have f(α)6=y06=f(β), and the continuity of fimplies f(α), f(β)∈Bd(y0, s). Then f(α), f(β)∈ V\{y0}, which is arcwise connected. Therefore, in the first case, we can find a continuous mapping h=hα,β : [α, β]→V\ {y0}satisfying h(α) = f(α) and h(β) = f(β). Define the mapping g:I→Yas follows: g(t) = f(t) if t∈I \f−1(Bd(y0, s)), g(t) = hα,β(t) if tbelongs to one of the intervals (α, β) making up f−1(Bd(y0, s)), g(t) = f(β) if t∈[0, β)⊂f−1(Bd(y0, s)), and g(t) = f(α) if t∈(α, 1] ⊂f−1(Bd(y0, s)). It is evident that gis continuous and g(t)6=y0for all t∈I. Then g6∈ PY. Now, the triangle inequality and the fact s < r/2 yield d(g(t), f(t)) < r for all t∈I, so g∈Bρ(f, r). This contradiction proves (c). (d) Fix α∈(0,1) and a ball Bρ(f, ε)⊂C(I, I2). Define f0:= (1 −ε 2)f. Trivially, f0∈C(I, I2). Moreover, ρ(f, f0) = sup t∈I kf(t)−(1 −ε 2)f(t)k1=ε 2sup t∈I kf(t)k1≤ε 2< ε, so f0∈Bρ(f, ε). Take g∈Bρ(f0, αρ(f, f0)). Then d(g(t), f0(t)) ≤αρ(f, f0)≤ αε/2 for all t∈Iand, by the triangle inequality, kg(t)k1≤αρ(f, f0) + kf0(t)k1≤αε 2+ 1 −ε 2<1. Therefore g(I)6=I2, so P ∩ Bρ(f0, αρ(f, f0)) = ∅. This had to be shown.  8 BERNAL-GONZ ´ ALEZ, CALDER ´ ON-MORENO AND PRADO-BASSAS Remark 3.2. Of course, the condition in (c) above is fulfilled if Y=I2, but in this case the conclusion of (d) is stronger than that of (c). Notice that some assumption on Yis really needed in order that P0 Y=∅. For instance, for the unit circle Y=S1={(x, y)∈R2:x2+y2}, which clearly does not satisfy the mentioned condition, we have that P0 S16=∅. Indeed, it is not difficult to show that for the mapping f:t∈I7→ (cos(4πt),sin(4πt)) ∈S1 (which travels S1twice in the same direction) one has Bρ(f, 1/2) ⊂ PS1. Remark 3.3. The last theorem yields that, topologically speaking, the Peano class is very small. Another property of PY–easy to see and not related to the size– is that it is arcwise connected. The next statement tells us that, if we endow C(I, I2) with the semigroup structure given by coordinatewise multiplication, then Phas a chance to be considered large. Theorem 3.4. The set Pis semigroupable. Proof. Fix any sequence (an) with a1< a2<··· < an<· · · → 1. Let a0:= 0. According to Lemma 2.7, we can find for every n∈Na mapping fn∈CS([an−1, an], I2) with fn(an−1) = (1,1) = fn(an). Let us extend continuously fnto Iby defining fn(t) = (1,1) if t∈I\[an−1, an]. Then fn∈ P and, trivially, every power fm nstill belongs to P. Consider the subsemigroup Ggenerated by {fn}n≥1. Given Φ ∈G, there exist p∈N, {i1<··· < ip} ⊂ Nand {m1,...,mp} ⊂ Nsatisfying Φ = fm1 i1···fmp ip. Since I2⊃Φ(I)⊃Φ([aip−1, aip]) = fm ip([aip−1, aip]) = I2, we obtain Φ(I) = I2or, that is the same, G⊂ P. All that must be proved is that Gis infinitely generated. Assume, by way of contradiction, that there are finitely many elements of Ggenerating it. Taking into account the structure of Gand the fact that Gis commutative, there would be p∈Nsuch that each Φ ∈Gcan be written as Φ = fm1 1···fmp p, for some m1,...,mp∈ {0,1,2,...}depending on Φ. But taking Φ = fp+1, the previous equality is not possible, because fj(t) = (1,1) for all t∈[ap, ap+1] and all j= 1,...,p. This is the desired contradiction.  Remark 3.5. If Pis considered as a subset of the additive group (C(I, R2),+), then it is not very likely for the sum of two given mappings in Pto stay still in P. Nevertheless, we can say at least the following: given N∈N, there are functions f1,...,fN∈ P such that f1+···+fN∈ P. Indeed, for each i∈ {1, ..., N}take as fithe mapping Φ provided in Lemma 2.7 with Y=I2, [a, b] = [i−1 N,i N], u= (0,0) = v, extended as (0,0) to the remaining of I. But one cannot find a sequence {fn}∞ n=1 ⊂ P such that Pn≥1fnconverges uniformly to any function because, if this is were the case, one would have limn→∞ supt∈Ikfn(t)k1= 0, which is plainly not possible since supt∈Ikfn(t)k1= 1 for every n. We do not know whether SPACE-FILLING CURVES: TOPOLOGICAL AND ALGEBRAIC STRUCTURE 9 there is a sequence {fn}∞ n=1 ⊂ P such that Pn≥1fnconverges pointwise to a function f∈ P. 4. The family of space-filling curves Throughout this section we shall deal with the algebraic size of the set SF, viewed as a subset of C(I, R2). Recall that, under the distance given by (1), C(I, R2) is an F-space, that is, a completely metrizable topological vector space. In fact, it is a Banach space under the norm k · k := ρ(·,0). Concerning elementary topological properties, the set SF is clearly nonclosed in C(I, R2): if we take f∈ P then each fn:= (1/n)f∈ SF (n≥1) and fn→(0,0) 6∈ SF. Moreover, SF0=∅. Indeed, if ϕ∈C(I, R2) and ε > 0 are given, with ϕ(t) = (g(t), h(t)), then from the uniform continuity of gand hone obtains an N∈Nsuch that |g(t)−g(u)|< ε/2 and |h(t)−h(u)|< ε/2 whenever t, u ∈[i−1 N,i N] (i= 1,...,N). If we define eg, e h:I→Ras the polygonal functions joining successively the points ( i N, g(i N)) (i= 1, ..., N) and, respectively, the points ( i N, h(i N)) (i= 1, ..., N), then the mapping eϕ(t) := (eg(t),e h(t)) satisfies ρ(eϕ, ϕ)< ε and eϕ6∈ SF, so SF does not contain any ρ-ball. If ϕ∈C(I, R2) then ϕ(I) is compact, hence bounded and closed. Then ϕ(I) = ϕ(I)0∪∂ϕ(I). Therefore, according to Definition 1.1 and the final paragraph of Section 2, we have that ϕ∈ SF if and only if λ(∂ϕ(I)) = 0 and (ϕ(I))06=∅. We saw in Section 1 how Osgood’s example provided a λ-space-filling curve ψthat is not space-filling. In this case, we have even that (ϕ(I))0=∅; indeed, a continuous injective mapping I→R2cannot fill in a square, see [26]. In view of this, the following concept is in order. Definition 4.1. A continuous mapping ϕ:I→R2is said to be a topologically space-filling curve provided that (ϕ(I))06=∅. The family of all these mappings will be denoted by T SF. It is evident that SF ⊂ T SF ⊂ λ-SF := {λ-space-filling curves}. Moreover, both inclusions are strict. Indeed, for the second one we can appeal Osgood’s example, while for the first one we can construct on [0,1/3] a curve filling I2, and on [2/3,1] an Osgood-type curve that is disjoint with I2, and then to joint them along [1/3,2/3] by a segment so as to built a T SF mapping. In the following theorems it is shown that, in some algebraic senses, our family SF can be thought as “large”. Theorem 4.2. The family SF is spaceable in C(I, R2). In particular, it is maximal lineable. Proof. Fix again any sequence (an) with a1< a2<···< an<· · · → 1. 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Zamfirescu, Porosity in Convexity, Real Analysis Exchange 15 (1989), 424–436. Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Sevilla, Apdo. 1160, Avenida Reina Mercedes, Sevilla, 41080, Spain. E-mail address:[email protected], [email protected] and [email protected]