Asymptotically convex Banach spaces and The Index of Rotundity Problem

The Index of Rotundity Problem asks whether a Banach space which admits equivalent renormings with index of rotundity as small as desired also admits an equivalent rotund renorming. In this paper we continue the ongoing search for a negative answer to this question by making use of a new concept: asymptotically convex Banach spaces. Some applications to The Approximation Hyperplane Series Property are given. Subjclass : [2000] Primary 46B20, 46B10.


Introduction
Since the concept of rotund Banach space appeared many characterizations of it have been provided.This concept motivated, among other things, the creation of an index to measure it, the so called index of rotundity.Definition 1.1.Let X be a real or complex Banach space.The index of rotundity of X is defined as The reader can quickly observe that a normed space X is rotund if and only if ζ X = 0.In [3,Theorem 7.12] it is shown an example of a Banach space which is not rotund under any equivalent renorming.Such example triggered the search for a Banach space not isomorphic to a rotund space but which can be equivalently renormed to decrease its index of rotundity as much as desired.This is the birth of the "Index of Rotundity Problem", a well-known problem among the Banach space geometers.
Question 1.2 (The Index of Rotundity Problem).Let X be a real or complex Banach space.Assume that for every h > 0 there exists an equivalent renorming k•k h on X such that ζ (X,k•k h ) ≤ h.Does then there exist an equivalent norm k•k 0 on X such that ζ (X,k•k 0 ) = 0?
We refer the reader to [3] for a wide perspective on rotund renormings and the index of rotundity.In the upcoming sections we continue the ongoing search for a negative answer to this question.In the last section we provide some applications to The Approximation Hyperplane Series Property.

Asymptotically convex Banach spaces
This section is devoted to describe the class of Banach spaces we will be working with.Definition 2.1.Let X be a real or complex Banach space.Consider a closed subspace Y of X and two numbers ε ∈ (0, 1] and δ ∈ [0, 1).We say that X is (ε, δ)-asymptotically convex in the direction of Examples of Banach spaces asymptotically convex follow.
In this situation, X is (1, 0)-asymptotically convex in the direction of 2 can be used as a base to construct infinite dimensional asymptotically convex Banach spaces.This construction will actually show us that every Banach space is isomorphic to an asymptotically convex Banach space.
The norm of the space X in Example 2.2 can actually be described as: where x, y ∈ R.
Theorem 2.3.Let Y and Z be Banach spaces such that Z is rotund.The vector space X := Y × Z endowed with the norm where y ∈ Y and z ∈ Z, is a Banach space (1, 0)-asymptotically convex in the direction of Y × {0}.

Proof.
Let [(y 1 , z 1 ) , (y 2 , z 2 )] be a non-trivial segment contained in S X \ S Y ×{0} .Observe that the three points (ky 1 k , kz 1 k), (ky 2 k , kz 2 k), and ³°°°y 1 +y 2 2 °°°, °°z 1 +z 2 2 °°´a re in the unit sphere of the 2-dimensional Banach space constructed in Example 2.2.More precisely, the three previous points must be in the positive part of that unit sphere.On the other hand, notice also that °°°y Therefore, the only possibility is that kz due to the fact that Z is rotund.But this contradicts the fact that Let X be a real or complex Banach space.Then X can be equivalently renormed to become (1, 0)-asymptotically convex in the direction of a closed subspace.

Proof.
Consider Z to be any complemented subspace of X which admits an equivalent rotund renorming (for instance, any finite dimensional subspace).Assume that Z is already endowed with that equivalent rotund renorming.Let Y be the topological complement of Z in X. Finally consider the norm given by Equation (2.1) on Y × Z. 2

Decreasing the index of rotundity
The next two results are aimed at showing that a certain type of asymptotically convex Banach space can be renormed in order to decrease the index of rotundity as much as desired.
Theorem 3.1.Let X be a real or complex Banach space (ε, δ)-asymptotically convex in the direction of a closed subspace Y .Consider the equivalent norm |•| on X given by |x| := q kxk 2 + dist 2 (x, Y ), for all x ∈ X. Then: Notice that Therefore, the previous two inequalities must indeed be equalities.In other words: • It must happen that • It must also be verified that which means that there exists α > 0 such that kx We have now all the necessary tools to prove the assertions (1) and (2) of the theorem: 2. Suppose now that [x 1 , x 2 ] ∩ Y =.Then kx 1 k = kx 2 k and therefore In the first place ´, In the second place ´.For all n ∈ N: As a consequence (and taking into consideration that kx Let X be a real or complex Banach space.Assume that there exist 0 < ε ≤ 1 and a rotund closed subspace Y of X such that X is (ε, 0)-asymptotically convex in the direction of Y .For every h ∈ (0, 2] there exists an equivalent norm |

Proof.
Denote by |•| 0 to the original norm on X.For every n ∈ N we define the equivalent norm |•| n on X given by for all x ∈ X.By applying Theorem 3.1 inductively, we deduce that n∈N converges to 0, we can find m ∈ N large enough so that We will conclude this section by showing the existence of Banach spaces with index of rotundity 2 under any equivalent renorming.
Let ω 1 denote the first uncountable ordinal.The space of all bounded real-valued functions on [0, ω 1 ] will be denoted by ∞ (0, ω 1 ), which becomes a Banach space endowed with the sup norm.The subspace of ∞ (0, ω 1 ) composed of those functions with countable support is denoted by m 0 .In accordance to [3,Theorem 7.12], m 0 endowed with any (non-necessarily equivalent) norm has a subspace which is linearly isometric to m 0 endowed with the sup norm.The reader may notice that the index of rotundity of m 0 endowed with the sup norm is 2; therefore the index of rotundity of m 0 endowed with any norm is also 2. As a consequence, if Γ is any uncountable set, then ∞ (Γ) has index of rotundity 2 under any equivalent renorming.

Applications to The Approximate Hyperplane Series Property
In [1] the authors find that the Property β of Lindenstrauss (see [4]) is a sufficient condition for a certain pair of Banach spaces to enjoy the Bishop-Phelps-Bollobás Property.In an attempt to provide a characterization of the Bishop-Phelps-Bollobás Property, the authors of [1] came up with a new property called The Approximation Hyperplane Series Property, which we will introduce afterwards.The characterization they gave was the following: Theorem 4.2 (Acosta, Aron, Garc ía, and Maestre, 2008).Let Y be a real or complex Banach space.The following conditions are equivalent: 1.The pair ( 1 , Y ) has the Bishop-Phelps-Bollobás Property.

Y satisfies the Approximation Hyperplane Series Property.
We will proceed now to define the Approximation Hyperplane Series Property.Within the next lines we will let M denote the set of all functions γ : [0, 2] → h 0, 1 2 i such that: • γ (ε) > 0 for all 0 < ε ≤ 2.
Given a real or complex Banach space X, γ ∈ M, and ε ∈ [0, 2], we define the set HS (X, γ, ε) in the following way: k=1 α k is a convex series, and for all A ⊆ N with P k∈A α k > 1−γ (ε) and for all (z k ) k∈A with co {z k : k ∈ A} ⊂ S X , there exists k 0 ∈ A verifying that kz k 0 − x k 0 k ≥ ε.
Definition 4.3 (Acosta, Aron, Garc ía, Maestre, 2008).Let X be a real or complex Banach space: 1.The Modulus of Hyperplane Series Approximation of X with respect to γ ∈ M is the function 2. X has the Approximation Hyperplane Series Property if there exists γ X ∈ M such that η X (ε, γ X ) > 0 for all 0 < ε ≤ 2.
The geometrical interpretation of the Approximation Hyperplane Series Property is that somehow a convex series of a sequence in the unit sphere can be approximated by a unit ball supporting hyperplane.The reader may notice that the previous definition is not exactly the way the authors of [1] defined the Approximation Hyperplane Series Property.Actually, the modulus of hyperplane series approximation was later introduced in a different manuscript (see [2]).In [2] the authors prove the following result.Theorem 4.4 (Acosta, Aron, Garc ía-Pacheco, 2010).Let X be a real or complex Banach space.
We remind the reader that δ X denotes the modulus of convexity (or modulus of Clarkson) of the Banach space X.Therefore, Theorem 4.4 shows that the modulus of convexity and the modulus of hyperplane series approximation are somehow related.Another result proved in [2] follows.
Theorem 4.5 (Acosta, Aron, Garc ía-Pacheco, 2010).Let X be a real or complex Banach space.If X admits an equivalent rotund renorming, then X admits an equivalent rotund renorming failing the Approximation Hyperplane Series Property.
We will finish this section and the paper by providing a version of the previous result for asymptotically convex Banach spaces, making use of Theorem 4.4.
Theorem 4.6.Let X be a real or complex Banach space.Assume that there exist 0 < ε ≤ 1 and a closed subspace Y of X such that X is (ε, 0)asymptotically convex in the direction of Y .If there exists ε 0 ∈ [3ζ Y , 2] such that δ Y (ε 0 ) = 0, then there exists an equivalent norm on X so that X is (ε, 0)-asymptotically convex in the direction of Y but lacks the Approximation Hyperplane Series Property.

Proof.
In the first place, following a similar argument as in the proof of Corollary 3.

Theorem 4 . 1 (
Acosta, Aron, Garc ía, and Maestre, 2008).Let Y be a real or complex Banach space.If Y satisfies Property β, then the pair (X, Y ) has the Bishop-Phelps-Bollobás Property for every Banach space X.