NORMAL STRUCTURE AND MODULUS

OF U-CONVEXITY IN BANACH SPACES

Ji Gao

Let X be a normed linear space, and let

S(X) = {x ∈ X: ∥x∥ = 1}

be the unit sphere of X. Brodskii and Milman [1] introduced the following geometric concept:

Definition 1. A bounded, convex subset K of a Banach space X is said to have normal structure if every convex subset H of K that contains more than one point contains a point x_{0} ∈ H, such that

sup{∥x_{0} − y∥: y ∈ H} < d(H),

where d(H) = sup{∥x−y∥: x, y ∈ H} denotes the diameter of H. A Banach space X is said to have normal structure if every bounded, convex subset of X has normal structure. A Banach space X is said to have w-normal structure if every weakly compact set K in X that contains more than one point has normal structure. It is clear that if X is reflexive, then the normal structure and the w-normal structure coincide.

Kirk [6] proved that if a weakly compact subset K of X has normal structure, then every nonexpansive mapping on K has a fixed point. It is well known that uniformly convex Banach spaces [2], uniformly smooth spaces [8], the Banach spaces which are uniformly convex in every direction [9], the nearly uniformly convex spaces [3], U-spaces [4], and the Banach spaces with J(X) < \frac{3}{2}, where J(X) = sup\{∥y − x∥ ∧ ∥y + x∥: x, y ∈ S(X)\} [4], have normal structure.

Gao and Lau [4] showed that if X fails to have w-normal structure, then X satisfies geometric conditions given in the following lemma:

Lemma. Let X be a Banach space without w-normal structure, then for any 0 < ε < 1 there exist x_{1}, x_{2}, x_{3}, in S(X) satisfying

(i) x_{2} − x_{3} = ax_{1} with |a − 1| < ε;

(ii) |∥x_{1} − x_{2}∥ − 1| < ε, |∥x_{3} − (−x_{1})∥ − 1| < ε;

(iii) \frac{1}{2}∥x_{1} + x_{2}∥ > 1 − ε, \frac{1}{2}∥x_{3} + (−x_{1})∥ > 1 − ε.

In this paper we give a new class of Banach spaces that have normal structure.

For x ∈ S(X), let ∇_{x} denote the set of norm 1 supporting functionals of S(X) at x.

Definition 2. Let X be a Banach space. The modulus of u-convexity of X is defined as u(ε) = inf \{1 − \frac{1}{2}∥x + y∥: x, y ∈ S(X), and ⟨f, x − y⟩ ≥ ε for some f ∈ ∇_{x}}, 0 ≤ ε ≤ 2.

Proposition 1. u(ε) is an increasing function mapping [0, 2] into [0, 1].

Proof. Let ε_{1} < ε_{2}. Then ⟨f, x − y⟩ ≥ ε_{2} implies ⟨f, x − y⟩ ≥ ε_{1}. So, u(ε_{1}) ≤ u(ε_{2}). □

Let δ(ε) = inf \{1 − \frac{1}{2}∥x + y∥: x, y ∈ S(X), ∥x − y∥ ≥ ε\}, 0 ≤ ε ≤ 2, be the modulus of convexity of X.

Proposition 2. u(ε) ≥ δ(ε).

Proof. ∀ε > 0, and f ∈ ∇_{x}, ⟨f, x − y⟩ ≥ ε implies ∥x − y∥ ≥ ε. □

In [7], Lau introduced the following notion in order to study the Chebyshev subset of X:

Definition 3. A Banach space X is called a U-space if for every ε > 0 there exists a δ > 0 such that for x, y ∈ S(X)

∥\frac{x + y}{2} ∥ > 1 − δ ⇒ ⟨f, y⟩ > 1 − ε, ∀f ∈ ∇_{x}.

Theorem 1. Let X be a Banach space, and let u(ε) be the modulus of u-convexity of X, then X is a U-space if and only if u(ε) > 0 for every ε > 0.

Proof. “⇒” Let ε > 0. By Definition 3, there exists a δ > 0, such that if x, y ∈ S(X), and ⟨f, y⟩ ≤ 1 − ε for some f ∈ ∇_{x}, then we have ∥\frac{1}{2}(x + y)∥ ≤ 1 − δ. That is, if x, y ∈ S(X) and ⟨f, x − y⟩ ≥ ε for some f ∈ ∇_{x}, then 1 − ∥\frac{1}{2}(x + y)∥ ≥ δ.

NORMAL STRUCTURE AND MODULUS OF U-CONVEXITY 197

So, by Definition 2, u(ε) = inf \{1− \frac{1}{2}∥x+y∥: x, y ∈ S(X), and ⟨f, x−y⟩ ≥ ε, for some f ∈ ∇_{x}} > δ > 0.

“⇐” Let ε > 0. Take δ such that 0 < δ ≤ u(ε). Let x, y ∈ S(X). If there exists a f ∈ ∇_{x} such that ⟨f, y⟩ ≤ 1 − ε, i.e., if ⟨f, x − y⟩ ≥ ε for some f ∈ ∇_{x}, then we have 1 − ∥\frac{1}{2}(x + y)∥ ≥ u(ε) by Definition 2. Thus, ∥\frac{1}{2}(x + y)∥ ≤ 1 − u(ε) ≤ 1 − δ.

This means that for x, y ∈ S(X)

∥\frac{x + y}{2} ∥ > 1 − δ ⇒ ⟨f, y⟩ > 1 − ε, ∀f ∈ ∇_{x}.

So, X is a U-space by Definition 3. □

A normed linear space is called uniformly nonsquare [5] if there exists a δ > 0 such that for x, y ∈ S(X) either ∥\frac{1}{2}(x+y)∥ ≤ 1−δ or ∥\frac{1}{2}(x−y)∥ ≤ 1−δ.

Theorem 2. For any Banach space X, if u(1) > 0, then X is uniformly nonsquare.

Proof. u(1) = inf \{1 − \frac{1}{2}∥x + y∥: x, y ∈ S(X), and ⟨f, x − y⟩ ≥ 1, for some f ∈ ∇_{x}\} > 0 implies that if x, y ∈ S(X), and 1 − \frac{1}{2}∥x − y∥ < u(1), then we have ⟨f, x − y⟩ < 1, ∀f ∈ ∇_{x}; i.e., if x, y ∈ S(X), and \frac{1}{2}∥x + y∥ > 1 − u(1), then we have ⟨f, x − y⟩ < 1, ∀f ∈ ∇_{x}.

If X is not a uniformly nonsquare space, then for above u(1) > 0 there exist x, y ∈ S(X) such that both \frac{1}{2}∥x+y∥ > 1−u(1) and \frac{1}{2}∥x−y∥ > 1−u(1). So, both ⟨f, x − y⟩ < 1 and ⟨f, x + y⟩ < 1, ∀f ∈ ∇_{x}. Hence, both ⟨f, y⟩ > 0 and ⟨f, −y⟩ > 0. This is a contradiction. □

Theorem 3. For any Banach space X, if there exists a δ > 0 such that u(\frac{1}{2} − δ) > 0, then X has normal structure, and therefore X has the fixed point property.

Proof. The inequality u(\frac{1}{2} − δ) > 0 implies u(1) > 0, which in turn implies that X is uniformly nonsquare by Theorem 2, and this implies that X is reflexive [5]. So the normal structure and w-normal structure coincide.

Since u(\frac{1}{2} − δ) > 0, we have that ⟨f, x − y⟩ < \frac{1}{2} − δ, for every f ∈ ∇_{x} and for every x, y ∈ S(X) such that \frac{1}{2}∥x + y∥ > 1 − u(\frac{1}{2} − δ).

Suppose that X does not have normal structure. Then X does not have w-normal structure either. Then there exist points x_{1}, x_{2}, x_{3} ∈ S(X) satisfying the three conditions in Lemma with respect to ε = min \{u(\frac{1}{2} − δ), δ\}.

Let f be any supporting functional of x_{1} ∈ S(X), i.e., f ∈ ∇_{x_{1}}.

198 JI GAO

Since

\frac{1}{2}∥x_{1} + x_{2}∥ > 1 − ε > 1 − u(\frac{1}{2} − δ),
\frac{1}{2}∥x_{1} − x_{3}∥ > 1 − ε > 1 − u(\frac{1}{2} − δ),
∥x_{2} − x_{3}∥ ≤ 1 + ε ≤ 1 + δ,

we have

⟨f, x_{1} − x_{2}⟩ < \frac{1}{2} − δ,
⟨f, x_{1} + x_{3}⟩ < \frac{1}{2} − δ,
⟨f, x_{2} − x_{3}⟩ ≤ ∥x_{2} − x_{3}∥ < 1 + δ.

Therefore

2 = 2⟨f, x_{1}⟩ = ⟨f, x_{1} − x_{2}⟩ + ⟨f, x_{1} + x_{3}⟩ + ⟨f, x_{2} − x_{3}⟩
< \frac{1}{2} − δ + \frac{1}{2} − δ + 1 + δ = 2 − δ.

Since δ > 0, this is impossible. □

From Theorem 1 and Theorem 3 we have the following corollaries: Corollary 1 ([4]). U-spaces have normal structure.

Since uniformly convex spaces and uniformly smooth spaces are both U-spaces [4], we have Corollary 2 ([2], [8]). Both uniformly convex spaces and uniformly smooth spaces have normal structure.

Banach space X is said to have uniform normal structure if there exists 0 < c < 1 such that for any subset K as in Definition 1, there exists x ∈ K such that

sup{∥x − y∥: y ∈ K} < c diam K.

By using the ultraproduct, Theorem 3 can be refined to the case of uniform normal structure. Theorem 4. For any Banach space X, if there exists a δ, \frac{1}{2} > δ > 0, such that u(\frac{1}{2} − δ) > 0, then X has uniform normal structure.

References

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[2] M. M. Day, Normed linear spaces, second edition, Springer-Verlag, New York-Heidel-berg-Berlin, 1973.

[3] D. van Dulst and B. Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), Banach Spaces Theory and its Applications, Proceedings, Bucharest, Lecture Notes in Math., vol. 991, Springer-Verlag, 1983, pp. 35–43.

[4] J. Gao and K. S. Lau, On two classes of Banach spaces with normal structure, Studia Math. 99 (1991), no. 1, 41–56.

[5] R. C. James, Uniformly nonsquare Banach spaces, Ann. Math. 80 (1964), 542–550.

[6] W. A. Kirk, A fixed point theorem for mappings which do not increase distance, Amer. Math. Monthly 72 (1965), 1004–1006.

[7] K. S. Lau, Best approximation by closed sets in Banach spaces, J. Approx. Theory 23 (1978), 29–36.

[8] B. Turret, A dual view of a theorem of Baillon, Lecture Notes in Pure and Appl. Math., vol. 80, Marcel Dekker, 1982, pp. 279–286.

[9] V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. 87 (1971), 5–33.