SOME MODULI AND CONSTANTS RELATED TO METRIC FIXED POINT THEORY

enrique llorens fuster

0 Introduction

The classical modulus of convexity introduced by J.A. Clarkson in 1936 to define uniformly convex spaces is at the origin of a great number of moduli defined since then [A-U 88].

Indeed, there are a lot of quantitative descriptions of geometrical properties of Banach spaces. The most common way for creating these descriptions, is to define a real function (a ”modulus”) depending on the Banach space under consideration, and from this define a suitable constant or coefficient closely related to this function. The moduli and/or the constants are attempts to get a better understanding about two things:

• The shape of the unit ball of a space, and

• The hidden relations between weak and strong convergence of sequences.

One might well ask: Are there too many moduli for these purposes? Maybe! In part this is because many of these moduli involve very difficult computations, and, often there are intricate links between them. Moreover, it is not unusual to find some moduli defined in (seemingly) different ways, depending on the preferences of the writer.

The aim of this Appendix is to give only a brief summary of a few of these properties that are in some ways related to Metric Fixed Point Theory. It is not a comprehensive list of the geometrical properties of Banach spaces that have been described using moduli and/or constants. Among other things, such a list would be too long, and have many unavoidable overlaps.

Further, similar coefficients, for example those which were given in ([M 84]), have not been included as independent items. We have only concerned ourselves with coefficients and moduli defined for general Banach spaces. For example, we have not listed those defined in [Kh 87] or [J-L 96] for Banach spaces with the Schauder finite dimensional decomposition. For the sake of brevity, we have also considered only those constants depending on the weak topology, and of course, on the norm topology of the space. The author apologizes in advance for any omissions.

The excellent survey by Prus [Pr 96], as well as the chapter in this volume on the Geometrical Background for Metric Fixed Point Theory, has helped guide the preparation of this summary.

We shall assume throughout this paper that (X, ∥ · ∥) is a Banach space with dimension at least two. We will use B_{X} and S_{X} to stand for the unit ball and the unit sphere of X, respectively. Wherever possible we have retained the original notation used for the various moduli and constants, however a few exceptions were necessary to avoid confusion.

PART I. MODULI AND RELATED PROPERTIES

1 Clarkson’s Modulus of Convexity, [Cl 36]

It is the function δ_{X} : [0, 2] ⟶ [0, 1] given by

δ_{X}(ε) := inf \{1 − ∥\frac{1}{2}(x + y)∥ : x, y ∈ B_{X}, ∥x − y∥ ≥ ε\} .

[figure]

Related coefficient Characteristic of convexity of (X, ∥.∥)

ε_{0}(X) := sup{ε ∈ [0, 2] : δ_{X}(ε) = 0}.

[figure]

Geometrical properties in terms of this modulus and/or this coefficient

1. Definition 1 The Banach space (X, ∥ · ∥) is uniformly convex (UC) whenever δ_{X}(ε) > 0 for 0 < ε ≤ 2, or equivalently if ε_{0}(X) = 0.

2. The Banach space (X, ∥ · ∥) is strictly convex (SC) if and only if δ_{X}(2) = 1.

3. The Banach space (X, ∥ · ∥) is uniformly nonsquare if and only if ε_{0}(X) < 2.

4. James, Enflo.

For a Banach space (X, ∥ · ∥) the following statements are equivalent

a) X is superreflexive.

b) X has an equivalent uniformly nonsquare norm.

c) X has an equivalent uniformly convex norm.

5. ([G 70]) If δ_{X}(1) > 0 then (X, ∥ · ∥) (is reflexive) and has (uniform) normal structure.

6. (Gao-Lau [G-L 91]) If δ_{X} (\frac{3}{2}) > \frac{1}{4} then (X, ∥ · ∥) has uniform normal structure.

7. (Gao-Lau [G-L 91]) If δ_{X} (ε) ≥ \frac{1}{6}ε for some ε ∈ (0, \frac{3}{2}] then (X, ∥ · ∥) has uniform normal structure.

8. ([Pr 91]) If δ_{X}(ε) > δ_{h}(ε) := max \{0, \frac{ε−1}{2}\} then (X, ∥ · ∥) has uniform normal structure.

[figure]

Facts about this modulus and/or this coefficient

1. In the above definition ’B_{X}’ and ’≥’ can be replaced respectively by ’S_{X}’ and ’=’. (For a proof see, for instance, [Dn 76]).

2. δ_{X} is a monotone increasing function on [0, 2].

3. If Y is a closed subspace of X, δ_{Y} (ε) ≥ δ_{X}(ε) for ε ∈ [0, 2].

4. For x, y, p ∈ X, R > 0, and r ∈ [0, 2R]

\begin{matrix}∥x − p∥ ≤ R \\ ∥y − p∥ ≤ R \\ ∥x − y∥ ≥ r \end{matrix} \} ⇒ ∥p − \frac{1}{2}(x + y)∥ ≤ (1 − δ_{X} (\frac{r}{R})) R.

In particular,

\begin{matrix}∥x∥ ≤ R \\ ∥y∥ ≤ R \end{matrix} \} ⇒ ∥\frac{1}{2}(x + y)∥ ≤ (1 − δ_{X} (\frac{∥x − y∥}{R})) R.

5. For all u, v ∈ B_{X} and c ∈ [0, 1],

2 min{c, 1 − c}δ_{X}(∥u − v∥) ≤ 1 − ∥cu + (1 − c)v∥.

(See [G-K 90], p. 111).

6. If (H, ∥.∥) is a Hilbert space,

δ_{H}(ε) := \frac{2 − \sqrt{4 − ε^{2}}}{2} .

7. (Nordlander, [N 60]) The highest possible value of δ_{X} is attained in Hilbert spaces, that is, for any Banach space (X, ∥ · ∥)

δ_{X}(ε) ≤ \frac{2 − \sqrt{4 − ε^{2}}}{2} .

8. δ_{X} is continuous in [0, 2). (See [G-K 90] or [Ds 84] for a proof). (Also at ε = 2 if (X, ∥ · ∥) is (UC), [A-U 88]).

9. There are examples for which δ_{X} is not a convex function. See Goebel-Kirk, [G-K 90], p.60.

10. (Hanner, [H 56]. See also Prus, [Pr 96]).

Let (Ω, μ) be a measure space such that μ takes at least two different values. For p ≥ 2,

δ_{L^{p}(Ω,μ)}(ε) = 1 − (1 − (ε/2)^{p})^{\frac{1}{p}} .

For 1 < p < 2 then

(1 − δ_{L^{p}(Ω,μ)}(ε) + \frac{ε}{2})^{p} + |1 − δ_{L^{p}(Ω,μ)}(ε) − \frac{ε}{2}|^{p} = 2.

This shows in particular that the spaces L^{p}(Ω, μ) are (UC) whenever 1 < p < ∞. Clearly this covers the case of the spaces ℓ_{p}.

11.

lim_{ε→2^{−}} δ_{X}(ε) = 1 − \frac{1}{2}ε_{0}(X).

See [G-K 90].

12. For all ε ∈ [ε_{0}(X), 2],

δ_{X} (2(1 − δ_{X}(ε))) = 1 − \frac{1}{2}ε_{0}(X).

See [G-K 90].

13. See [Ds 84], p.125.

\frac{δ_{X}(ε_{1})}{ε_{1}} ≤ \frac{δ_{X}(ε_{2})}{ε_{2}}

whenever 0 < ε_{1} < ε_{2} ≤ 2.

14. (D.J. Downing and B. Turret, [D-T 83]).

If μ is a measure,

ε_{0} (L^{p}(μ, X)) = max{ε_{0}(ℓ^{p}), ε_{0}(X)}.

15. Let λ ≥ 1 and let X_{λ} denote the space obtained by renorming the Hilbert space (ℓ_{2}, ∥ · ∥)

by means of

∥x∥_{λ} := max{λ^{−1}∥x∥, ∥x∥_{∞}}.

Then

ε_{0}(X_{λ}) = \{ \begin{matrix}2(λ^{2} − 1)^{\frac{1}{2}} & λ ≤ \sqrt{2} \\ 2 & λ ≥ \sqrt{2}. \end{matrix}

16. ([Pr 96]) Let E be the space ℝ^{2} endowed with the norm ∥(x, y)∥ := max{|x|, |y|, |x − y|}.

We have

δ_{E}(ε) = max \{0, \frac{1}{2}(ε − 1)\} ⇒ ε_{0}(E) = 1.

17. ([G-K-T 74]) Let (X, ∥ · ∥) be a uniformly convex Banach space, and let C be a convex, bounded and closed subset of X. If T : C → C is uniformly lipschitzian with constant k and k is less than the (unique) solution of the equation

h (1 − δ_{X} (\frac{1}{h})) = 1,

then T has a fixed point in C.

18. ([D-T 83]) Stability of condition ε_{0}(X) < 1.

Let X be a Banach space and let X_{1} := (X, ∥.∥_{1}) and X_{2} := (X, ∥.∥_{2}), where ∥.∥_{1} and ∥.∥_{2} are two equivalent norms on X satisfying for α, β > 0,

α∥x∥_{1} ≤ ∥x∥_{2} ≤ β∥x∥_{1}

for all x ∈ X.

If ε_{0}(X_{1}) < 1 and if

k (1 − δ_{X_{1}} (\frac{1}{k} )) < 1,

where k := \frac{β}{α}, then ε_{0}(X_{2}) < 1.

Equivalent statement:

[D-T 83], Th. 6. Let (X, ∥ · ∥) be a Banach space with ε_{0}(X) < 1. Let Y be a Banach space isomorphic to X. If k is the unique solution of the equation

k (1 − δ_{X} (\frac{1}{k} )) = 1,

and d(X, Y) < k, then ε_{0}(Y) < 1.

19. ([K-K 95]) If ε_{0}(X) = 1 then every nonempty bounded closed convex subset of X has the fixed point property for asymptotically nonexpansive mappings.

2 Lovaglia local modulus of convexity, [Lv 55]

For x ∈ S_{X}, the modulus of convexity of X at the point x, is the function

δ_{X}(x, ε) := inf \{1 − ∥\frac{1}{2}(x + y)∥ : x, y ∈ B_{X}, ∥x − y∥ ≥ ε\} .

[figure]

Related coefficient

Characteristic of convexity of (X, ∥ · ∥) at x ∈ S_{X}

ε_{0}(X, x) := sup{ε ∈ [0, 2] : δ_{X}(x, ε) = 0}.
[figure]

Geometrical property in terms of this modulus and/or this coefficient

Definition 2 Lovaglia [Lv 55]. The Banach space (X, ∥ · ∥) is locally uniformly rotund (LUR) whenever δ_{X}(x, ε) > 0 for all x ∈ S_{X} and ε ∈ (0, 2].

[figure]

Separation of this property

1. Obviously, every (UC) Banach space is (LUR). The converse is not true:

Example 1 M.A. Smith, 1978. [Sm 78]

Let ℓ_{1} be endowed with the norm

∥x∥ := (∥x∥_{1}^{2} + ∥x∥_{2}^{2})^{\frac{1}{2}} .

Then for all x ∈ ℓ_{1}

∥x∥_{1} ≤ ∥x∥ ≤ \sqrt{2}∥x∥_{1}

and (ℓ_{1}, ∥.∥) is (LUR) but it does not admit a uniformly convex norm (otherwise it would be a reflexive Banach space).

2. (LUR) Banach spaces are strictly convex. The converse is not true again.

Example 2 M.A. Smith, 1978. [Sm 78]

Let ℓ_{2} be endowed with the norm

∥x∥_{w} := (∥x∥_{S}^{2} + ∥T x∥_{2}^{2})^{\frac{1}{2}} .

where ∥x∥_{S} := max{|x_{1}|, ∥(0, x_{2}, …)∥_{2}}, and

T (x_{1}, x_{2}, x_{3}, ...) := (x_{1}, \frac{1}{2}x_{2}, \frac{1}{3}x_{3}, ...).

Then the norm ∥.∥_{w} is equivalent to ∥.∥_{2} but (ℓ_{2}, ∥.∥_{w}) is reflexive (SC) but not (LUR).

3. Example 3 See [N-S-W 87]. Consider the classical space c_{0} endowed with Day’s norm

∥x∥ := sup [∑_{1}^{∞} \frac{1}{2^{j}} (x(α_{j}))^{2}]^{\frac{1}{2}} ,

where the supremum is taken over all permutations α of the positive integers.

One has that Z := (c_{0}, ∥.∥) is (LUR) and hence (SC). Nevertheless, the well known mapping T : B_{Z}^{+} → B_{Z}^{+} given by

T (x_{1}, x_{2}, ...) := (1, x_{1}, x_{2}, ...)

is ∥.∥-nonexpansive and lacks fixed points. Thus, (LUR)⇏(FPP).

3 Smulian’s modulus of weak uniform rotundity

For x^{∗} ∈ S_{X^{∗}}, the modulus of convexity of X with respect to x^{∗}, is the function given by δ_{X}(x^{∗}, .) : [0, 2] ⟶ [0, 1]

δ_{X}(x^{∗}, ε) := inf (\{1\} ∪ \{1 − ∥\frac{1}{2}(x + y)∥ : x, y ∈ S_{X}, |x^{∗}(x − y)| ≥ ε\}) .

The reason for specifically including 1 in the set whose infimum defines this modulus is to avoid the following particular situation. When x^{∗} is a non-norm attaining functional there are no points x, y in S_{X} such that |x^{∗}(x − y)| ≥ 2. So δ_{X}(x^{∗}, 2) would not be well defined.

[figure]

Geometrical property in terms of this modulus and/or this coefficient

Definition 3 Smulian, 1939. The Banach space (X, ∥ · ∥) is weakly uniformly rotund (WUR) whenever δ_{X}(x^{∗}, ε) > 0 for all x^{∗} ∈ S_{X} and ε ∈ (0, 2].

[figure]

Separation of this property

1. Neither of the conditions (LUR) and (WUR) implies the other. (See [Sm 78]).

2. Every (UC) Banach space is (WUR). See, for example, the book of R.E. Megginson, p.465, [Mg 98].

The converse is not true: The space (ℓ_{2}, ∥.∥_{w}) given in the M.A. Smith’s paper [Sm 78] is (WUR) but not (UC).

3. (WUR) Banach spaces are strictly convex. See again the book of R.E. Megginson, p.465, [Mg 98] for a proof.

The converse is not true.

Example 4 M.A. Smith, [Sm 78]. Recall the equivalent norm defined on c_{0} by M.M. Day: For u in c_{0} enumerate the support of u as (n_{k}) in such a way that |u(n_{k})| ≥ |u_{n_{k+1}}|. Define Du ∈ ℓ_{2} by

Du(n) := \{ \begin{matrix}\frac{u(n_{k})}{2^{k}} & n = n_{k} \\ 0 & otherwise \end{matrix}

and define |∥u|∥ = ∥Du∥_{2}. For x ∈ ℓ_{2} let

∥x∥_{L} := |∥(\frac{1}{2}∥x∥_{2}, x_{1}, x_{2}, x_{2}, ..., \overbrace{x_{j}, x_{j}, ...x_{j}}^{j}, ...)|∥

Then (ℓ_{2}, ∥.∥_{L}) is (LUR) and hence rotund, but not (WUR).

4 V.L. Smulian modulus of weak^{∗} uniform rotundity

For x ∈ S_{X}, the modulus of convexity of X^{∗} with respect to x, is the function δ_{X}(x, .) : [0, 2] ⟶ [0, 1] given by the formula

δ_{X^{∗}}(x, ε) := inf \{1 − ∥\frac{1}{2}(x^{∗} + y^{∗})∥ : x^{∗}, y^{∗} ∈ S_{X^{∗}}, |(x^{∗} − y^{∗})(x)| ≥ ε\}

[figure]

Geometrical property in terms of this modulus and/or this coefficient

Definition 4 V.L. Šmulian, 1939. The Banach space (X^{∗}, ∥.∥) is weak^{∗} uniformly rotund (W^{∗}UR) whenever δ_{X^{∗}}(x, ε) > 0 for all x ∈ S_{X} and ε ∈ (0, 2].

Megginson’s book [Mg 98], on page 466 reads

It should be noted that some sources say that if X is a normed space such that X^{∗} satisfies the above definition of weak^{∗} uniform rotundity, then it is X instead of X^{∗} which is called weak^{∗} uniformly rotund.

1. ([Mg 98], p. 466)

For dual Banach spaces, (WUR)⇒(W^{∗}UR)⇒ (SC).

5 Lovagia modulus of weak local uniform rotundity

For x ∈ S_{X} and x ∈ S_{X^{∗}}, the modulus of convexity of X at x with respect to x^{∗}, is the function δ_{X}(x, x^{∗}, .) : [0, 2] ⟶ [0, 1]

δ_{X}(x, x^{∗}, ε) := inf (\{1\} ∪ \{1 − ∥\frac{1}{2}(x + y)∥ : y ∈ S_{X}, |x^{∗}(x − y)| ≥ ε\}) .

[figure]

Geometrical property in terms of this modulus and/or this coefficient

Definition 5 Lovaglia, 1955. The Banach space (X, ∥ · ∥) is weakly locally uniformly rotund (WLUR) whenever δ_{X}(x, x^{∗}, ε) > 0 for all x^{∗} ∈ S_{X} and ε ∈ (0, 2].

[figure]

Separation of this property

1. (LUR)⇒ (WLUR). (See [Mg 98]) The converse is not true: The space (ℓ_{2}, ∥.∥_{w}) given in [Sm 78] is (WLUR) but not (LUR).

2. (WUR)⇒ (WLUR)⇒ (SC). None of the implications of this statement is reversible. M.A. Smith’s example (ℓ_{2}, ∥.∥_{L}) is (WLUR) but not (WUR).

Example 5 M.A. Smith, [Sm 78]. For all x in ℓ_{2} define

∥x∥_{F} := |x_{1}| + ∥(0, x_{2}, x_{3}, ...)∥_{2}

and

∥x∥_{A} := (∥x∥_{F}^{2} + ∥(\frac{1}{2}x_{2}, \frac{1}{3}x_{3}, ...)∥_{2}^{2})^{\frac{1}{2}} .

Then (ℓ_{2}, ∥.∥_{A}) is a reflexive (SC) not (WLUR) Banach space.

6 Lindenstrauss Modulus of Smoothness, [Ld 63]

It is the function ρ_{X} : [0, ∞) ⟶ ℝ given by

ρ_{X}(t) := sup \{\frac{1}{2}(∥x + ty∥ + ∥x − ty∥) − 1 : x, y ∈ B_{X}\} .

[figure]

Related coefficient

ρ_{0}(X) := lim_{t→0^{+}} \frac{ρ_{X}(t)}{t}.

[figure]

Geometrical properties in terms of this modulus and/or this coefficient

1. Definition 6 The Banach (X, ∥·∥) is uniformly smooth (US) whenever ρ_{0}(X) = 0.

2. ρ_{0}(X) < \frac{1}{2} ⇒ X is superreflexive and has (UNS). (See [G-K 90] p.70-71 for detailed references about this result).

[figure]

Facts about this modulus and/or this coefficient

1. For each positive t,

max{0, t − 1} ≤ ρ_{X}(t) ≤ t.

(See, for instance, [Mg 98] for a proof).

2. Alternative formula for ρ_{X}:

ρ_{X}(t) = sup \{\frac{1}{2}(∥x + y∥ + ∥x − y∥) − 1 : x ∈ B_{X}, ∥y∥ ≤ t\} .

3. For a Hilbert space H ρ_{H}(t) = \sqrt{1 + t^{2}} − 1 for every t ≥ 0. ρ_{X}(t) ≥ ρ_{H}(t) for t ≥ 0.

4. Let (Ω, μ) be a measure space such that μ takes at least two different values. For 1 < p ≤ 2

then

ρ_{L^{p}(Ω,μ)}(t) = (1 + t^{p})^{\frac{1}{p}} − 1,

and for p > 2, then

ρ_{L^{p}(Ω,μ)}(t) = (\frac{(1 + t)^{p} + |1 − t|^{p}}{2} )^{\frac{1}{p}} − 1

for every t ≥ 0. (See [Ld 63]). Clearly this covers the case of the spaces ℓ_{p}.

5. For every Banach space (X, ∥ · ∥) , the modulus of smoothness of (X, ∥ · ∥) is a convex continuous function.

6. Lindenstrauss formulae (see [Ld 63]).

ρ_{X^{∗}}(t) = sup \{\frac{1}{2}εt − δ_{X}(ε) : 0 ≤ ε ≤ 2\} .

ρ_{X}(t) = sup \{\frac{1}{2}εt − δ_{X^{∗}}(ε) : 0 ≤ ε ≤ 2\} .

Thus, ρ_{0}(X^{∗}) = \frac{1}{2}ε_{0}(X) and ρ_{0}(X) = \frac{1}{2}ε_{0}(X^{∗}). Consequently, (X, ∥ · ∥) is (UC) if and only if its dual space is (US) and (X, ∥ · ∥) is (US) if and only if its dual space is (UC).

7. ρ_{0}(X) = 0 if and only if for each positive η there exits a positive ε (depending on η) such

that, if x ∈ S_{X}, y ∈ X and ∥x − y∥ ≤ ε, then

∥x + y∥ ≥ ∥x∥ + ∥y∥ − η∥x − y∥.

(See [Dy 73], p. 147).

7 Directional modulus of rotundity

For a given non zero z ∈ X, the modulus of convexity of X in the direction of z, is the function δ_{X}(→ z, ·) : [0, 2] ⟶ [0, 1]

δ_{X}(→ z, ε) := inf \{1 − ∥\frac{1}{2}(x + y)∥ : x, y ∈ S_{X}, ∥x − y∥ ≥ ε, ∃λ ∈ ℝ s.t. x − y = λz\} .

[figure]

Related coefficient

ε_{0,z}(X) := sup{ε : δ_{X}(→ z, ε) = 0}.
[figure]

Geometrical properties in terms of this modulus and/or this coefficient

1. Definition 7 The Banach space (X, ∥·∥) is uniformly convex in every direction (UCED) whenever δ_{X}(→ z, ε) > 0 for all z ∈ X\{0} and ε ∈ (0, 2].

2. ([G 70]) If ε_{0,z}(X) < 1 for all z ∈ X, z ≠ 0, then (X, ∥ · ∥) has normal structure.

3. ([D-J-S 71], [Zi 71]) (UCED) ⇒ (W NS).

[figure]

Separation of (UCED)

1. (WUR)⇒ (UCED)⇒ (SC).

The converse implications are not true.

The above example (ℓ_{2}, ∥.∥_{A}) is a (UCED) Banach space which is not (WUR). On the other hand, (ℓ_{2}, ∥.∥_{L}) is (SC) but not (UCED).

[figure: Facts about this modulus](See Prus’ chapter in this book).

1. For a Hilbert space H, δ_{H}(→ z, ε) = 1 − \sqrt{1 − (\frac{ε}{2})^{2}}, for every z ∈ S_{H} and ε ∈ [0, 2].

2. δ_{X}(→ z, ·) need not be a convex function.

3. Every separable Banach space admits an equivalent norm with respect to which it is (UCED).

8 Gurarii modulus of convexity, [Gu 67]

It is defined by the formula

β_{X}(ε) := inf {1 − inf_{t∈[0,1]} ∥tx + (1 − t)y∥ : x, y ∈ S_{X}, ∥x − y∥ = ε} .

[figure: Facts about this modulus](See Sánchez and Ullán [S-U 98]).

1. For any ε ∈ [0, 2], δ_{X}(ε) ≤ β_{X}(ε) ≤ 2δ_{X}(ε).

2. (X, ∥ · ∥) is (SC) if and only if β_{X}(2) = 1.

3. (X, ∥ · ∥) is uniformly nonsquare if and only if there exists ε ∈ (0, 2) such that β_{X}(ε) > 0.

4. (X, ∥ · ∥) is uniformly convex if and only β_{X}(ε) > 0 for all ε ∈ (0, 2].

5. β_{X}(ε) is a continuous function at [0, 2). Continuity may fail at ε = 2. Nevertheless, if (X, ∥ · ∥) is a uniformly convex Banach space then β_{X} is continuous at ε = 2.

6. If (X, ∥ · ∥) is an inner product space, β_{X}(ε) = δ_{X}(ε).

7. If the inequality β_{X}(ε) ≥ 1 − \sqrt{1 − \frac{ε^{2}}{4}} holds in a normed space (X, ∥ · ∥) for every ε ∈ (0, 2] then (X, ∥ · ∥) is an inner product space.

8. β_{X}(ε) = inf {1 − inf_{t∈[0,1]} ∥tx + (1 − t)y∥ : x, y ∈ B_{X}, ∥x − y∥ = ε} .

9. β_{X} is nondecreasing on [0, 2].

10. β_{X}(ε) = inf {1 − inf_{t∈[0,1]} ∥tx + (1 − t)y∥ : x, y ∈ B_{X}, ∥x − y∥ ≥ ε} .

11. For all r ∈ [0, 1], β_{X}(rε) ≤ rβ_{X}(ε).

12. The function ε ↦ \frac{β_{X}(ε)}{ε} is non-decreasing in (0, 2].

13. β_{X} is strictly increasing in [ε_{0}(X), 2].

14. If (X, ∥ · ∥) is strictly convex and β_{X} is continuous at ε = 2 then (X, ∥ · ∥) is uniformly nonsquare.

15. For p ≥ 1 and (X, ∥ · ∥) is the Banach space ℓ_{p} or L^{p} then for all ε ∈ [0, 2],

β_{X}(ε) ≤ 1 − (1 − (\frac{1}{2})^{p})^{\frac{1}{p}} .

16. For p ≥ 2 and for all ε ∈ [0, 2],

β_{ℓ_{p}}(ε) = β_{L^{p}}(ε) = 1 − (1 − (\frac{1}{2})^{p})^{\frac{1}{p}} .

9 Milman Modulus of Convexity

It is the function d_{X} : [0, ∞) ⟶ [0, ∞) given by (see [Mi 71]) d_{X}(ε) := inf {max{∥x + εy∥, ∥x − εy∥ : x, y ∈ S_{X}, } − 1} .

[figure]

Main features of this modulus

1. Figiel’s formula [Fg 76] : For every ε ∈ [0, 2)

\frac{δ_{X}(ε)}{1 − δ_{X}(ε)} = d_{X} (\frac{ε}{2(1 − δ_{X}(ε))}) .

2. This formula shows that (X, ∥ · ∥) is (UC) if and only if d_{X}(ε) > 0 for all ε > 0.

10 Geremia-Sullivan Modulus of k-rotundity

For x_{1}, x_{2}, …, x_{k+1} ∈ X the value

V (x_{1}, x_{2}, …, x_{k+1}) := sup \{det ( \begin{matrix}1 & … & 1 \\ f_{1}(x_{1}) & … & f_{1}(x_{k+1}) \\ ⋮ & ⋯ & ⋮ \\ f_{k}(x_{1}) & … & f_{k}(x_{k+1}) \end{matrix} ) : f_{1}, …, f_{k} ∈ B_{X^{∗}} \}

is called k-volume of the set conv {x_{1}, …, x_{k+1}}.

The function

δ_{X}^{(k)}(ε) := inf \{1 − \frac{1}{k + 1}∥x_{1} + … + x_{k+1}∥ : x_{1}, …, x_{k+1} ∈ B_{X}, V (x_{1}, …, x_{k+1}) ≥ ε\}

is called the modulus of k-rotundity of (X, ∥.∥). Clearly this function is defined on the interval [0, μ_{k}(X)) where

μ_{k}(X) := sup{V (x_{1}, …, x_{k+1}) : x_{1}, …, x_{k+1} ∈ B_{X}}.

[figure]

Related coefficient Characteristic of k-convexity of (X, ∥ · ∥)

ε_{0}^{(k)}(X) := sup{ε > 0 : δ_{X}^{(k)}(ε) = 0}.

[figure]

Geometrical properties in terms of this modulus and/or this coefficient

1. Definition 8 Sullivan, [Su 79]. (See also Geremia-Sullivan, [G-S 81]). The Banach space (X, ∥ · ∥) is k-uniformly rotund (k-UR) whenever δ_{X}^{(k)}(ε) > 0 for ε ∈ (0, μ_{k}(X)). (The case k = 1 corresponds to Clarkson’s uniform convexity.)

2. ([B-S 83])If ε_{0}^{(k)}(X) < 1 then (X, ∥ · ∥) has (UNS).

[figure]

Separation of (k-UR) In [Ku 91] D. Kutzarova gave the following Example 6 Let X be the ℓ_{1} direct sum of the Banach spaces Y = ℝ^{1} and Z = ℓ_{2}. Then X

is 2-uniformly rotund. But ε_{0}(X) = 2. Of course, X is not uniformly convex.

[figure: Facts about this modulus and/or this coefficient](See Prus’ chapter in this book)

1. In the definition of δ_{X}^{k}(ε) ’≥ ε’ can be replaced by ’= ε’.

2. The function δ_{X}^{k}(·) is nondecreasing.

3. The function δ_{X}^{k}(·) is continuous on [0, μ_{k}(X)).

4. The function δ_{X}^{k}(·) is lipschitzian on each interval (a, b)where 0 < a < b < μ_{k}(X).

5. The function ε ↦ \frac{δ_{X}^{k}(ε^{k})}{ε} is nondecreasing.

6. (UC)⇒ (kUR) ⇒ ((k + 1)UR). In fact,

δ_{X}(ε) = δ_{X}^{(1)}(ε) ≤ δ_{X}^{(k)}(ε) ≤ δ_{X}^{(k+1)}(ε).

for all positive integer k and all suitable ε.

7. If X is infinite dimensional, as well as the Hilbert space H,

δ_{X}^{k}(ε) ≤ δ_{H}^{k} = 1 − (1 − \frac{k}{(k + 1)^{1+1/k}} ε^{2/k})^{\frac{1}{2}} .

8. ([B-S 83]) Let k ∈ ℕ. If ε_{0}^{(k)}(X) < 2^{k} then X is superreflexive.

11 Milman k-dimensional modulus of convexity

Notation: E_{k} is the collection of all (k + 1)-dimensional subspaces of X.

In the same paper of Milman there seems to have been defined the following modulus of k-dimensional convexity. (Here we are following the survey of Prus [Pr 96]).

Δ_{X}^{(k)}(ε) := inf_{x∈S_{X}} inf_{E∈E_{k}} sup {∥x + εy∥ − 1 : y ∈ S_{E}} .

[figure]

Geometrical property in terms of this modulus and/or this coefficient Definition 9 The Banach space (X, ∥ · ∥) is k-uniformly convex (k-UC for short) when-

ever Δ_{X}^{(k)}(ε) > 0 for ε > 0. Geremia and Sullivan for k = 2 and P.K. Lin ([Ln 88]) in the general case have shown that k-uniform convexity and k-uniform rotundity are equivalent properties.

[figure: Facts about this modulus](See Prus’ chapter in this book)

1. 0 = Δ_{X}^{(k)}(0) ≤ Δ_{X}^{(k)}(ε) for every ε ≥ 0.

2. The function ε ↦ \frac{Δ_{X}^{(k)}(ε)}{ε} is nondecreasing in (0, ∞).

3. Δ_{X}^{(k)}(t) ≤ Δ_{X}^{(k+1)}(t) for every t ≥ 0. Consequently (k − UC) ⇒ ((k + 1) − UC).

4. ([Fg 76])

Δ_{X}^{1} (\frac{ε}{2(1 − δ_{X}(ε))}) = \frac{δ_{X}(ε)}{1 − δ_{X}(ε)}.

12 Milman k-dimensional moduli of smoothness

Notation: E_{k} is the collection of all k-dimensional subspaces of X.

For each x ∈ S_{X} and t ≥ 0, the k-dimensional modulus of smoothness at x is given by

β_{X}(k, x, t) := sup_{E∈E_{k}} \{inf\{\frac{∥x − ty∥ + ∥x + ty∥}{2} − 1 : y ∈ S_{E}\}\} .

Milman’s k-dimensional modulus of smoothness is given by

β_{X}^{(k)}(t) := sup{β_{X}(k, x, t) : x ∈ S_{X}}.

[figure]

Geometrical property in terms of this modulus and/or this coefficient

Definition 10 A Banach space (X, ∥ · ∥) is called k-uniformly smooth, (k-US) for short, if

lim_{t→0^{+}} \frac{β_{X}^{(k)}(t)}{t} = 0.

[figure: Properties of this modulus](See Prus’ chapter in this book)

1. 0 = β_{X}^{(k)}(0) ≤ β_{X}^{(k)}(ε) for every ε ≥ 0.

2. The function ε ↦ \frac{β_{X}^{(k)}(ε)}{ε} is nondecreasing in (0, ∞).

3. β_{X}^{(k+1)}(t) ≤ β_{X}^{(k)}(t) for every t ≥ 0. Consequently (k − US) ⇒ ((k + 1) − US).

13 Partington modulus of ”UKK-ness”, [Pa 83]

It was defined by J.R. Partington in the proof of Theorem 1 of [Pa 83]. It is the function P_{X} : [0, K(X)) ⟶ [0, 1] given by

P_{X}(ε) := inf{1 − ∥x∥ : ∃x_{n} ∈ B_{X} (n = 1, …), x_{n} ⇀^{w} x, ∥x_{m} − x_{n}∥ ≥ ε, (m ≠ n)}.

Here K(X) = K(B_{X}) is the so called Kottman constant of the space (X, ∥ · ∥) , that is,

K(B_{X}) := sup {{inf ∥x_{m} − x_{n}∥ : m ≠ n} : x_{n} ∈ B_{X} (n = 1, ...)} .
[figure]

Main features of this modulus

1.

P_{ℓ_{p}}(ε) = 1 − (1 − \frac{ε^{p}}{2} )^{\frac{1}{p}}

for ε ∈ [0, 2^{\frac{1}{p}}). (See [D-L 92], or [B-F 93]).

14 Goebel-Sekowski modulus of noncompact convexity, [G-S 84]

Let us suppose that dim(X) = ∞. The modulus of noncompact convexity of a Banach space (X, ∥ · ∥) is the function Δ_{X} : [0, 2] → [0, 1] defined by

Δ_{X}(ε) := inf \{1 − dist (0, A) : A ⊂ B_{X}, A ≠ ∅, A = \overline{co} (A), α(A) ≥ ε\} .

Here α(A) := inf{r > 0 : A ⊂ B_{1} ∪ ... ∪ B_{k}, diam(B_{i}) < r} is the Kuratowski measure of noncompactness of the set A ⊂ X.

[figure]

Related coefficient

ε_{1}(X) := sup{ε ∈ [0, 2] : Δ_{X}(ε) = 0}.

is called Characteristic of noncompact convexity of (X, ∥ · ∥) .

[figure]

Geometrical property in terms of this modulus and/or this coefficient

1. Definition 11 A space for which ε_{1}(X) = 0 is said to be Δ-uniformly convex.

2. A Banach space is (NUC) if dim(X) < ∞ or dim(X) = ∞ and Δ_{X}(ε) > 0 for every ε ∈ (0, 2].

Recall that a Banach space is said to be nearly uniformly convex (NUC) if for any ε > 0 there exists δ > 0 such that for any sequence (x_{n}) in B_{X} with sep((x_{n})) > ε one has that

dist(0, conv ({x_{n}})) < 1 − δ.

Here sep((x_{n})) := inf{∥x_{n} − x_{m}∥ : m ≠ n}.

[figure]

Separation of this property

Example 7 Let ℓ_{1}^{n} (ℓ_{∞}^{n}) be the space ℝ^{n} endowed with the norm ∥·∥_{1}, (∥·∥_{∞}). We consider the Day spaces

\begin{matrix}D_{1} := (ℓ_{1}^{1} × ℓ_{1}^{2} × … × ℓ_{1}^{n} × …)_{ℓ_{2}} \\ D_{∞} := (ℓ_{∞}^{1} × ℓ_{∞}^{2} × … × ℓ_{∞}^{n} × …)_{ℓ_{2}}. \end{matrix}

Then one has (see [G-S 84]) that for all ε ∈ [0, 2]

Δ_{D_{1}}(ε) = Δ_{D_{∞}}(ε) = δ_{H}(ε) = 1 − (1 − \frac{ε^{2}}{4} )^{\frac{1}{2}} .

Thus D_{1} and D_{∞} are Δ-uniformly convex. However, the spaces D_{1}, D_{∞} are not uniformly convex, nor even superreflexive. Thus δ_{D_{1}}(ε) = δ_{D_{∞}}(ε) ≡ 0 even under equivalent renormings.

The spaces D_{1} and D_{∞} do not have uniform normal structure although they have normal structure.

[figure]

Facts about this modulus and/or this coefficient

1. Δ_{X}(ε) ≥ δ_{X}(ε) for every ε ∈ [0, 2].

2. For any positive integer k, and 0 ≤ t < ε ≤ 2,

Δ_{X}(ε) ≥ δ_{X}^{(k)} ((\frac{t}{2})^{k}) .

See Kirk, 1988 ([Ki 88] ) for a proof.

3. ([B-F 93])

P_{X}(ε) ≥ Δ_{X} (\frac{ε}{K_{B_{X}}} )

where Δ_{X}(.) is the modulus of noncompact convexity.

4. For 1 < p < ∞,

Δ_{ℓ_{p}}(ε) = 1 − (1 − (\frac{ε}{2})^{p})^{\frac{1}{p}} .

One can see the proof in the joint work of Goebel and Sekowski [G-S 84].

5. If 2 ≤ p < ∞ and ε ∈ [0, 2]

Δ_{ℓ_{p}}(ε) = Δ_{L_{p}([0,1])}(ε) = δ_{L_{p}([0,1])}(ε).

See the paper of Banaś [Ba 90] for a proof. However, for 1 < p < 2 and ε ∈ (0, 2],

Δ_{ℓ_{p}}(ε) > Δ_{ℓ_{2}}(ε) = δ_{H}(ε) > δ_{ℓ_{p}}(ε).

6. Δ_{ℓ_{1}}(ε) ≡ 0. Note that lim_{p↓1} Δ_{ℓ_{p}}(ε) = \frac{ε}{2} ≠ Δ_{ℓ_{1}}(ε).

7. (See [Pr 96])

Δ_{X} = Δ_{ℓ_{p}}.

whenever X is the ℓ_{p} direct sum of a sequence (X_{n}) of finite dimensional Banach spaces.

8. The exact values of Δ_{L_{p}([0,1])} are not known if 1 < p < 2, although they are different from those of Δ_{ℓ_{p}}. For more details see the work of S. Prus (1994) [Pr 94].

9. For x ∈ ℓ_{p}, (1 ≤ p < ∞), we denote by x^{+} and x^{−} the vectors whose i component are respectively

x^{+}(i) := max\{x(i), 0\} = \frac{x(i) + |x(i)|}{2}, x^{−}(i) := max\{−x(i), 0\} = \frac{−x(i) + |x(i)|}{2}.

For any q ∈ [1, ∞), and for x ∈ ℓ_{p} we denote

\begin{matrix}∥x∥_{p,q} := (∥x^{+}∥_{p}^{q} + ∥x^{−}∥_{p}^{q})^{\frac{1}{q}} \\ ∥x∥_{p,∞} := max\{∥x^{+}∥_{p}, ∥x^{−}∥_{p}\}. \end{matrix}

It is easy to check that all these norms are equivalent to the usual norm in ℓ_{p}. The Banach spaces ℓ_{p,q} = (ℓ_{p}, ∥.∥_{p,q}) where introduced by Bynum. The modulus of noncompact convexity of the space ℓ_{p,1} is

Δ_{ℓ_{p,1}}(ε) = 1 − (1 − (\frac{ε}{2})^{p})^{\frac{1}{p}}

for any ε ∈ [0, 2]. (See [S 86]).

10. ([G-S 84]) If ε_{1}(X) < 1 then (X, ∥ · ∥) is reflexive. (Then, (NUC) Banach spaces are reflexive).

11. ([G-S 84]) If ε_{1}(X) < 1 then (X, ∥ · ∥) has normal structure.

12. ([S 86], [G-K 90]) Stability of condition ε_{1}(X) < 1.

Let X be a Banach space and let X_{1} := (X, ∥.∥_{1}) and X_{2} := (X, ∥.∥_{2}) where ∥.∥_{1} and ∥.∥_{2} are two equivalent norms on X satisfying for α, β > 0,

α∥x∥_{1} ≤ ∥x∥_{2} ≤ β∥x∥_{1}

for all x ∈ X. If k := \frac{β}{α} then

Δ_{X_{2}}(ε) ≥ 1 − k (1 − Δ_{X_{1}} (\frac{ε}{k} )) .

Consequently, if ε_{1}(X_{1}) < 1 and

k (1 − Δ_{X_{1}} (\frac{1}{k} )) < 1

then ε_{0}(X_{2}) < 1.

15 Banas modulus of noncompact convexity, [Ba 87]

Let us suppose that dim(X) = ∞. The modulus of noncompact convexity with respect to the Hausdorff measure of noncompactness of a Banach space (X, ∥·∥) is the function Δ_{X,χ} : [0, 1] ⟶ [0, 1] defined by

Δ_{X,χ}(ε) := inf {1 − dist (0, A) : A ⊂ B_{X}, A ≠ ∅, closed and convex, χ(A) ≥ ε} . Here χ(A) := inf{r > 0 : A ⊂ B_{1} ∪ ... ∪ B_{k}, B_{i} balls of radii smaller than r} is the Hausdorff measure of noncompactness of the set A ⊂ X.

[figure]

Related coefficient

ε_{χ}(X) := sup{ε ≥ 0 : Δ_{X,χ}(ε) = 0}.

is called Characteristic of noncompact convexity of (X, ∥ · ∥) associated to the Hausdorff measure of noncompactness.

[figure]

Geometrical properties in terms of this modulus and/or this coefficient

1. A space for which ε_{χ}(X) = 0 is said to be Δ_{χ}-uniformly convex.

2. ([Ba 87]) (X, ∥ · ∥) is (NUC) if and only if ε_{χ}(X) = 0.

3. ([Ba 87]) If ε_{χ}(X) < \frac{1}{2} then (X, ∥ · ∥) (is reflexive and) has normal structure.

4. ([G-J-L 94]) Banach spaces with nonstrict Opial condition and with ε_{χ}(X) < 1 have weakly normal structure. Recall that (X, ∥ · ∥) satisfies the nonstrict Opial condition provided that if a sequence in X is weakly convergent to x ∈ X then

lim inf_{n} ∥x_{n} − x∥ ≤ lim inf_{n} ∥x_{n} − y∥

for every y ∈ X.

[figure]

Facts about this modulus and/or this coefficient

1. ([Ba 87]) Δ_{X}(ε) ≤ Δ_{X,χ}(ε) ≤ Δ_{X}(2ε) for every ε ∈ [0, 1].

2. ([Ba 87]) The function Δ_{X,χ}(·) is continuous on the interval [0, 1).

3. ([Ba 87]) If X is a reflexive Banach space then (a) Δ_{X,χ} is a subhomogeneous function, that is Δ_{X,χ}(kε) ≤ kΔ_{X,χ}(ε) for any k, ε ∈ [0, 1].

(b) For any ε ∈ [0, 1], Δ_{X,χ}(ε) ≤ ε.

(c) The function Δ_{X,χ}(·) is strictly increasing on the interval [ε_{χ}(X), 1]. The function ε ↦ Δ_{X,χ}(ε)ε is nondecreasing on (0, 1] and Δ_{X,χ}(ε_{1} + ε_{2}) ≥ Δ_{X,χ}(ε_{1}) + Δ_{X,χ}(ε_{2}) whenever 0 ≤ ε_{1} + ε_{2} ≤ 1.

4. ([A-D-L 97], p. 89) If ε_{χ}(X) < 1 then X is reflexive.

5. ([Ba 90])

Δ_{ℓ_{p},χ}(ε) = 1 − (1 − ε^{p})^{\frac{1}{p}} .

6. ([Ba 87]) Stability of condition ε_{χ}(X) < \frac{1}{2}.

Let X be a Banach space with ε_{χ}(X) < \frac{1}{2}. Let B > 1 be such that

1 − \frac{1}{B} = Δ_{χ} (\frac{1}{2B} ) .

(which exists in view of the continuity of the function Δ_{χ}). If Y is another Banach space with d(X, Y) < B, then Δ_{χ}(Y) < \frac{1}{2}.

7. ([D-J 01]) Let (X, ∥·∥) be a Banach space, C a nonempty weakly compact subset of X and T : C → C and asymptotically regular mapping. Let h := sup \{t ≥ 1 : \frac{1}{t}Δ_{X,χ} (\frac{1}{t}) ≥ 1\}. If lim inf_{n} |T^{n}| < h, then T has a fixed point. (Here |T | denotes the (exact) Lipschitz constant of T on C).

16 Domínguez-López modulus of noncompact convexity, [D-L 92]

Let us suppose that dim(X) = ∞. The modulus of noncompact convexity with respect to the Istraţescu measure (or separation measure) of noncompactness of a Banach space (X, ∥ · ∥) is the function Δ_{X,β} : [0, β(B_{X})] ⟶ [0, 1] defined by

Δ_{X,β}(ε) := inf {1 − dist (0, A) : A ⊂ B_{X}, A ≠ ∅, closed and convex , β(A) ≥ ε} .

Here β(A) := sup{r > 0 : A has an infinite r separation}, where a r separation of A is a nonempty subset S ⊂ A such that ∥x − y∥ ≥ r for all x, y ∈ S, x ≠ y.

[figure]

Related coefficient

ε_{β}(X) := sup{ε ≥ 0 : Δ_{X,β}(ε) = 0}.

is called characteristic of noncompact convexity of (X, ∥ · ∥) associated to the separation measure of noncompactness.

[figure]

Geometrical properties in terms of this modulus and/or this coefficient

1. A space for which ε_{β}(X) = 0 is said to be Δ_{β}-uniformly convex.

2. ([A-D-L 97]) (X, ∥ · ∥) is (NUC) if and only if ε_{β}(X) = 0.

3. ([D-L 92]) If ε_{β}(X) < 1 then (X, ∥ · ∥) has normal structure.

4. ([G-J-L 94]) If Δ_{X,β}(1) ≠ 0 then (X, ∥ · ∥) has weak normal structure.

[figure]

Facts about this modulus and/or this coefficient

1. ([A-D-L 97], p.86) δ_{X}(ε) ≤ Δ_{X}(ε) ≤ Δ_{X,β}(ε) ≤ Δ_{X,χ}(ε) and consequently

ε_{0}(X) ≥ ε_{1}(X) ≥ ε_{β}(X) ≥ ε_{χ}(X).

2. ([A-D-L 97], p. 90. ) If ε_{β}(X) < 1 then X is reflexive.

3. ([D-L 92]) For 1 < p < ∞,

Δ_{ℓ_{p},β}(ε) = 1 − (1 − \frac{ε^{p}}{2} )^{\frac{1}{p}} .

4. ([A-D-L 97], p. 96)

Δ_{D_{1},β}(ε) = Δ_{D_{∞},β}(ε) = 1 − (1 − \frac{ε^{2}}{2} )^{\frac{1}{2}} .

So the spaces D_{1} and D_{∞} are (NUC) but fail to be k-UC for any k.

5. ([A-D-L 97], Remark 1.12, p. 93) For reflexive spaces, Partington’s modulus is identical to the modulus of noncompact convexity associated to β.

6. ([D-J 01]) Let (X, ∥ · ∥) be a Banach space, C a nonempty weakly compact subset of X and T : C → C and asymptotically regular mapping. If lim inf_{n} |T^{n}| < 1/(1 − Δ_{X,β}(1^{−}) then T has a fixed point. (Here |T | denotes the (exact) Lipschitz constant of T on C).

17 Opial’s modulus

It was defined by S. Prus in [Pr 92]. See also (Lin-Tan-Xu, [LTX 95]).

It is the function r_{X} : [0, ∞) ⟶ ℝ given by

r_{X}(c) := inf{lim inf ∥x + x_{n}∥ − 1 : ∥x∥ ≥ c, x_{n} ⇀^{w} 0, lim inf_{n} ∥x_{n}∥ ≥ 1}.

[figure]

Geometrical properties in terms of this modulus and/or this coefficient Definition 12 A Banach space (X, ∥ · ∥) is said to satisfy the uniform Opial property

if for any c > 0 there exists an r > 0 such that

1 + r ≤ lim inf_{n→∞} ∥x + x_{n}∥

for each x ∈ X with ∥x∥ ≥ c and each sequence (x_{n}) in X such that x_{n} ⇀^{w} 0 and lim inf ∥x_{n}∥ ≥ 1.

1. The space (X, ∥ · ∥) satisfies the uniform Opial property if and only if r_{X}(c) > 0 for all c > 0.

2. ([X 96] and [D 01]) r_{X}(c) ≥ 0 for all c ≥ 0 if and only if (X, ∥ · ∥) has the nonstrict Opial property.

3. ([X 97]) If r_{X}(c) > 0 for some c ∈ (0, 1), then (X, ∥ · ∥) has weakly normal structure.

[figure]

Main features of this modulus

1. For all c ≥ 0,

c − 1 ≤ r_{X}(c) ≤ c.

In particular, r_{X}(c) > 0 for all c > 1. (See [LTX 95]).

2. The function r_{X} is continuous on [0, ∞).

3. The function c ↦ \frac{1+r_{X}(c)}{c} is nondecreasing on (0, ∞). In fact we have

r_{X}(c_{1}) − r_{X}(c_{2}) ≤ (c_{2} − c_{1})\frac{1 + r_{X}(c_{1})}{c_{1}}

whenever 0 < c_{1} ≤ c_{2}. (See [LTX 95])

4. For 1 < p < ∞

r_{ℓ_{p}}(c) = (1 + c)^{\frac{1}{p}} − 1

(c ≥ 0). (See [LTX 95]).

5. If 1 < p < ∞ and 1 ≤ q < ∞ then for all c ≥ 0,

r_{ℓ_{p,q}}(c) = min \{(1 + c^{p})^{\frac{1}{p}} − 1, (1 + c^{q})^{\frac{1}{q}} − 1\} .

(See [X 96]).

6. ([Kc 99] See also [K-R 97] for a generalization) Let X be a Banach space with r_{X}(1) > 0 and with the nonstrict Opial property, C a nonempty weakly compact subset of X and T = {T_{t} : t ∈ G} an asymptotically regular semigroup with

σ(T ) = k < 1 + r_{X}(1).

Then there exists z ∈ C such that T_{t}(z) = z for all t ∈ G. Here σ(T ) := lim inf |T_{t}| and k := sup_{t} |T_{t}|, where |T_{t}| denotes the exact Lipschitz constant of T_{t} in C.

7. ([LTX 95]) Suppose C is a weakly compact convex subset of X. If (X, ∥ · ∥) satisfies the uniform Opial property and T : C → C is a mapping of asymptotically nonexpansive type ; that is, for every x ∈ C,

lim sup_{n} [sup{∥T^{n}(x) − T^{n}(y)∥ − ∥x − y∥ : y ∈ C}] ≤ 0.

If T^{N} is continuous for some integer N ≥ 1, then T has a fixed point.

8. ([D-J 01]) Let (X, ∥ · ∥) be a Banach space, C a nonempty weakly compact subset of X and T : C → C and asymptotically regular mapping. If lim inf_{n} |T^{n}| < 1 + r_{X}(1) then T has a fixed point.

18 Six moduli for the property (β) of Rolewicz

In a Banach space (X, ∥ · ∥) the drop D(x, B_{X}) defined by an element x ∈ X\B_{X} is the set conv ({x} ∪ B_{X}), and we write R_{x} := D(x, B_{X})\B_{X}.

Recall that (X, ∥ · ∥) is said to have the property (β) if for each ε > 0 there exists δ > 0 such that

1 < ∥x∥ < 1 + δ ⇒ μ(R_{x}) < ε

where μ is any of the three measures of noncompactness α, β, S. (S(A) stands for the separation measure of the set A, that is S(A) := sup{ε > 0 : ∃(x_{n}) in A with sep(x_{n}) > ε}.)

In [A-D-F 94] the authors defined three moduli as follows

R_{X} : [0, 2] ⟶ [0, 1]:

R_{X}(ε) := 1 − sup \{inf \{\frac{1}{2}∥x + x_{n}∥ : n ∈ ℕ\} : \{x_{n}\} ⊂ B_{X}, x ∈ B_{X}, α(\{x_{n}\}) ≥ ε\} ,

R′_{X} : [0, 2] ⟶ [0, 1]:

R′_{X}(ε) := 1 − sup \{inf \{\frac{1}{2}∥x + x_{n}∥ : n ∈ ℕ\} : \{x_{n}\} ⊂ B_{X}, x ∈ B_{X}, β(\{x_{n}\}) ≥ ε\} ,

R′′_{X} : [0, a] ⟶ [0, 1]:

R′′_{X}(ε) := 1 − sup \{inf \{\frac{1}{2}∥x + x_{n}∥ : n ∈ ℕ\} : \{x_{n}\} ⊂ B_{X}, x ∈ B_{X}, sep (\{x_{n}\}) ≥ ε\}

where a is a real number in the interval [1, 2) depending on (X, ∥ · ∥) . Moreover, the same authors in [A-D-F 97] defined P_{X,μ} : [0, μ(B_{X})) → [0, ∞) by

P_{X,μ}(ε) := inf{∥x∥ − 1 : x ∈ X, ∥x∥ > 1, μ(R_{x}) ≥ ε}.

[figure]

Related coefficients

R_{0}(X) := sup{ε ≥ 0 : R_{X}(ε) = 0}.
R′_{0}(X) := sup{ε ≥ 0 : R′_{X}(ε) = 0}.
R′′_{0}(X) := sup{ε ≥ 0 : R′′_{X}(ε) = 0}.
P_{0,μ}(X) := sup{ε ≥ 0 : P_{X,μ}(ε) = 0}.
[figure]

Geometric properties in terms of these moduli or constants

1. (X, ∥ · ∥) has the property (β) of Rolewicz if and only if either R_{0}(X) = 0, R′_{0}(X) = 0, or R′′_{0}(X) = 0.

2. P_{0,μ}(X) = 0 if and only if (X, ∥ · ∥) has property (β) of Rolewicz.

It is easy to prove that (UC) implies property (β) and property (β) implies NUC. Thus, this property lies between uniform and near uniform convexity.

3. If any of the coefficients R_{0}(X), R′_{0}(X), R′′_{0}(X) is less than one, then both X and X^{∗} are reflexive and have normal structure.

4. If P_{0,μ}(X) < \frac{1}{2}, then the spaces X and X^{∗} are reflexive and have normal structure.

[figure]

Main features of these moduli 1.

δ_{X} (\frac{ε}{2}) ≤ R′_{X}(ε) ≤ R_{X}(ε) ≤ R′′_{X}(ε) ≤ Δ_{X,S}(ε).

2.

ε_{S}(X) ≤ R′′_{0}(X) ≤ R_{0}(X) ≤ R′_{0}(X) ≤ 2ε_{0}(X).

3. If 1 < p < ∞, then

R′′_{ℓ_{p}}(ε) = 1 − \frac{1}{2} \{\frac{ε^{p}}{2} + ((1 − \frac{ε^{p}}{2} )^{\frac{1}{p}} + 1)^{p}\}^{\frac{1}{p}} .

4. If 1 < p < ∞, then

R′_{ℓ_{p}}(ε) = 1 − \frac{1}{2} \{(\frac{ε}{2})^{p} + ((1 − (\frac{ε}{2})^{p})^{\frac{1}{p}} + 1)^{p}\}^{\frac{1}{p}} .

5. R′′_{ℓ_{∞}}(ε) = R′_{ℓ_{∞}}(ε) = R_{ℓ_{∞}}(ε) = 0 for all ε ∈ [0, 2].

6. R′′_{ℓ_{1}}(ε) = R′_{ℓ_{1}}(ε) = R_{ℓ_{1}}(ε) = 0 for all ε ∈ [0, 2].

7. 0 ≤ P_{X,μ}(ε) ≤ \frac{ε}{μ(B_{X})−ε}. It follows that P_{X,μ} is continuous at 0.

8. If 1 < p < ∞ and 0 ≤ ε < 2^{\frac{1}{p}}, then

P_{ℓ_{p},S}(ε) = (\frac{2}{2 − ε^{p}} )^{1− \frac{1}{p}} − 1.

9. If 1 < p < ∞ and 0 ≤ ε < 2, then

P_{ℓ_{p},α}(ε) = P_{ℓ_{p},β} = (\frac{2^{p}}{2^{p} − ε^{p}} )^{1− \frac{1}{p}} − 1.

10. P_{c,μ}(ε) = P_{c_{0},μ}(ε) = P_{ℓ_{1},μ}(ε) = P_{ℓ^{∞},μ}(ε) = 0. for all ε ∈ [0, 2].

19 Domínguez modulus of (NUS), [Do 95]

It is the function given by

Γ_{X}(t) := sup \{inf \{\frac{∥x_{1} + tx_{n}∥ + ∥x_{1} − tx_{n}∥}{2} : n > 1\} : (x_{n}) basic sequence in B_{X}\} .

[figure]

Geometrical properties in terms of this modulus and/or this coefficient

1. If X is (US) then lim_{t→0} \frac{Γ_{X}(t)}{t} = 0.

2. (X, ∥ · ∥) is (NUS) if and only if X is reflexive and lim_{t→0} \frac{Γ_{X}(t)}{t} = 0.

[figure]

Main features of these modulus

1. ρ_{X}(t) ≥ Γ_{X}(t) for every t ∈ [0, 2].

2. For 1 < p < ∞, Γ_{ℓ_{p}}(t) = (1 + t^{p})^{\frac{1}{p}}.

Other properties of this modulus are listed below, among the ones of W CS(X).

20 The Modulus of squareness, a universal one

It was defined in the joint paper by Przeslawski and Yost [P-Y 95]. For more information about it see the joint paper by Benítez, Przeslawski, Yost [B-P-Y 98] which is the source of all this information. Given a normed space (X, ∥ · ∥) , one observes that for any x, y ∈ X with ∥y∥ < 1 < ∥x∥, there is a unique z = z(x, y) ∈ S_{X} ∩ [x, y], (where [x, y] is the line segment joining x and y. We put

ω(x, y) := \frac{∥x − z(x, y)∥}{∥x∥ − 1}

and define χ_{X} : [0, 1) ⟶ [1, ∞) by

χ_{X}(β) := sup{ω(x, y) : ∥y∥ ≤ β < 1 < ∥x∥}.
[figure]

Geometrical properties in terms of this modulus and/or this coefficient 1. (X, ∥ · ∥) is uniformly convex if and only if

lim_{β→1^{−}}(1 − β)χ_{X}(β) = 0.

2. (X, ∥ · ∥) is uniformly smooth if and only if χ′_{X}(0) = 0. 3.

χ_{X^{∗}}(β) = \frac{1}{χ_{X}^{−1} (\frac{1}{β} )}.

4. If χ_{X}(β) < \frac{1}{1−β} for some β, then (X, ∥ · ∥) has uniformly normal structure.

5. If a normed space (X, ∥ · ∥) is uniformly nonsquare then for each β ∈ (0, 1),

χ_{X}(β) < χ_{1}(β) := \frac{1 + β}{1 − β} .

Here χ_{p} is χ_{X} where X is l_{p}.

[figure]

Main features of this modulus

1. χ_{X}(β) = sup{χ_{M}(β) : M ⊂ X, dim(M) = 2}.

2. χ_{X}(·) is a strictly increasing and convex function.

3. χ_{X}(β) = χ_{2}(β) := \frac{1}{\sqrt{1−β^{2}}} whenever (X, ∥ · ∥) is an inner product space.

4. For any normed space (X, ∥ · ∥) containing ℓ_{1}(2)

χ_{X}(β) = χ_{1}(β).

5. For all normed space (X, ∥ · ∥)

χ_{X}(β) ≤ χ_{1}(β).

6.

χ_{X}(β) < χ_{1}(β)

everywhere on (0, 1) unless (X, ∥ · ∥) contains arbitrarily close copies of ℓ_{1}(2). 7.

χ′_{X}(β) < χ′_{1}(β)

almost everywhere on (0, 1). 8.

χ_{X}(β) > χ_{2}(β)

everywhere on (0, 1) unless (X, ∥ · ∥) is an inner product space. 9.

χ_{X}(γ) − χ_{X}(β) ≤ χ_{1}(γ) − χ_{1}(β)

whenever 0 ≤ β < γ < 1.

10. Fix δ, β ∈ (0, 1). If χ_{X}(β) > (1 − δβ)χ_{1}(β), then (X, ∥ · ∥) contains a two-dimensional subspace whose Banach -Mazur distance from ℓ_{1}(2) is less than \frac{1}{1−(1+β)δ}.

11. If X and Y are two isomorphic Banach spaces whose Banach -Mazur distance is less than

1 + 2δ^{2} for some δ ∈ [0, 1], then for all β ∈ (0, 1)

|χ_{X}(β) − χ_{Y} (β)| ≤ \frac{2(δ + δ^{2})}{(1 − β)^{2}}.

12. Let ρ_{X} the modulus of smoothness of a Banach space (X, ∥ · ∥) . Then, for all β ∈ (0, 1)

1 ≤ \frac{χ_{X}(β) − 1}{ρ_{X}(β)} ≤ \frac{2}{1 − β} .

and χ′_{X}(0) = ρ′_{X}(0).

21 Three further modulus for the Nearly Uniform Convexity

They were defined by J.M. Ayerbe and S. Francisco ([A-F 97(a)]) as follows.

D_{X,μ} : [0, μ(B_{X})) ⟶ [0, ∞) is the function given by

D_{X,μ}(ε) := inf{∥x∥ − 1 : x ∈ X\B_{X}, μ̃(R_{x}) ≥ ε}

where, for a bounded subset A of X, μ̃(A) := sup{μ(C) : C = conv (C), C ⊂ A} and μ ∈ {α, χ, β}.

[figure]

Related coefficient The coefficients of noncompact convexity of (X, ∥ · ∥) corresponding to this modulus are the

numbers

D_{0,μ}(X) := sup{ε ≥ 0 : D_{X,μ}(ε) = 0}.
[figure]

Geometrical properties in terms of this modulus and/or this coefficient

1. The Banach space (X, ∥ · ∥) is (NUC) if and only if D_{0,μ}(X) = 0.

2. If D_{0,β}(X) < \frac{1}{2}, then the space (X, ∥ · ∥) is reflexive and has normal structure. [figure: Main features of this modulus](See [A-F 97(a)])

1. D_{X,μ} is nondecreasing in [0, μ(B_{X})), and D_{X,μ}(0) = 0.

2. For every ε ∈ [0, \frac{1}{2}μ(B_{X})) we have

0 ≤ D_{X,μ}(ε) ≤ \frac{ε}{μ(B_{X}) − ε}.

3. D_{X,μ} is continuous at ε = 0.

4. For every ε ∈ [0, μ(B_{X})) we have

D_{X,μ}(ε) < \frac{2ε}{μ(B_{X})}.

5.

ε_{β}(X) ≤ 2D_{0,β}(X) ≤ 4ε_{β}(X).

6. Let H be a separable and infinite dimensional Hilbert space. Then

\begin{matrix}D_{H,α}(ε) = \frac{2ε^{2}}{4−ε^{2}} & ε ∈ [0, 2) \\ D_{H,χ}(ε) = \frac{2ε^{2}}{1−ε^{2}} & ε ∈ [0, 1) \\ D_{H,β}(ε) = \frac{2ε^{2}}{2−ε^{2}} & ε ∈ [0, \sqrt{2}). \end{matrix}

22 A further modulus for the Uniform Convexity

It was defined by S. Francisco ([F 97]) as follows.

D_{X} : [0, ∞) ⟶ [0, ∞) is the function given by

D_{X}(ε) := inf{∥x∥ − 1 : x ∈ X\B_{X}, diam(R_{x}) ≥ ε}.

[figure]

Related coefficient The coefficient of convexity of (X, ∥ · ∥) corresponding to this modulus is the number

D_{0}(X) := sup{ε ≥ 0 : D_{X}(ε) = 0}.

[figure]

Geometrical properties in terms of this modulus and/or this coefficient

1. The Banach space (X, ∥ · ∥) is (UC) if and only if D_{0}(X) = 0.

2. If D_{0}(X) < \frac{1}{2}, then the space (X, ∥ · ∥) is reflexive and has normal structure.

[figure: Main features of this modulus](See [F 97] and [A-F 97(b)]).

1. D_{X} is nondecreasing on [0, ∞), and D_{X}(0) = 0.

2. For every ε ∈ [0, 1] we have 0 ≤ D_{X}(ε) ≤ \frac{ε}{2−ε}. Hence, if ε ∈ (0, 1), then D_{X}(ε) < ε.

3. D_{X} is continuous at ε = 0. 4.

ε_{0}(X) ≤ 2D_{0}(X) ≤ 8ε_{0}(X).

5. Let H be a separable Hilbert space. Then

D_{H}(ε) = \{ \begin{matrix}\frac{2}{\sqrt{4−ε^{2}}} − 1 & ε ∈ [0, \sqrt{3}] \\ \sqrt{1 + ε^{2}} − 1 & ε ∈ [\sqrt{3}, ∞). \end{matrix}

6. If (X, ∥ · ∥) is a uniformly convex Banach space, then D_{X} is strictly increasing on [0, ∞).

23 Modulus of uniform ”nonoctahedralness”

It was defined by A. Jiménez-Melado in [J 99]. Denote by δ̃(X) the supremum of the set of numbers ε ∈ [0, 2] for which there exist points x_{1}, x_{2}, x_{3} in B_{X} with min{∥x_{i} − x_{j}∥: i ≠ j} ≥ ε. Define the function δ̃_{X} : [0, δ̃(X)) → [0, 1] by

δ̃_{X}(ε) = inf \{1 − \frac{1}{3}∥x_{1} + x_{2} + x_{3}∥ : x_{i} ∈ B_{X}, i = 1, 2, 3, and min\{∥x_{i} − x_{j}∥: i ≠ j\} ≥ ε\} .

[figure]

Related coefficient with this modulus ε̃_{0}(X) := sup{ε ∈ [0, δ̃(X)): δ̃_{X}(ε) = 0}.

[figure]

Geometrical properties in terms of this modulus and/or this coefficient

1. ([G-L-M 01]) ε̃_{0}(X) < 1 ⇒ X has (UNS).

[figure: Main features of this modulus](See [J 99])

1. For any ε ∈ [0, δ̃(X)).

2. ε̃_{0}(X) ≠ ε_{0}(X)

In some cases, the inequality is strict.

3. Consider the classical real sequence space ℓ_{2} endowed with its usual euclidean norm ∥ · ∥.

Let | · | be a norm on ℓ_{2} such that

∥x∥ ≤ |x| ≤ b∥x∥ (x ∈ ℓ_{2})

for some b ≥ 1 and let X = (ℓ_{2}, | · |). Then ε̃_{0}(X) < 2 for b < \sqrt{\frac{7}{3}}.

4. Now consider the space E_{β} := (ℓ_{2}, | · |_{β}), where |x|_{β} = max{∥x∥, β∥x∥_{∞}}. Since ∥x∥ ≤ |x|_{β} ≤ β∥x∥_{2} for all x ∈ ℓ_{2} then ε̃_{0}(E_{β}) < 2 for β < \sqrt{\frac{7}{3}}. On the other hand, for β ≥ \sqrt{2} we have ε_{0}(E_{β}) = 2.

5. If (X, ∥ · ∥) is a Banach space with the (WORTH) property such that ε̃_{0}(X) < 2 then (X, ∥ · ∥) has the (WFPP). Recall that (X, ∥ · ∥) has the (WORTH) property if

lim_{n} |∥x_{n} − x∥ − ∥x_{n} + x∥| = 0

for all x ∈ X and for all weakly null sequences (x_{n}) in X.

6. Suppose that there exist δ > 0 and ε ∈ (0, 2) such that for all points x, y, z ∈ S_{X} with

∥(1/3)(x + y + z)∥ > 1 − δ we have

∥y − z∥dist(x, [y, z]) < ε.

Then ε̃_{0}(X) < 2. (Here [y, z] is the affine span of {y, z}.)

PART II. LIST OF COEFFICIENTS

24 Jung constant

It is the oldest of this list. It was defined by Jung in a work published in 1901 [Ju 01]. We will follow the Amir’s paper [Am 85] to list some important facts about it.

For a bounded subset A of X, and a subset Y of X, we denote by r_{Y} (A) the relative Chebyshev radius of A with respect to Y, that is,

r_{Y} (A) := inf {sup{∥x − y∥ : x ∈ A} : y ∈ Y } .

The Jung constant of (X, ∥ · ∥) is

J(X) := sup{2r_{X}(A) : A ⊂ (X), diam(A) = 1}.

Besides the Jung constant J one may study also the ”self-Jung” constant

J_{s}(X) := sup{2r_{A}(A) : A ⊂ X, diam(A) = 1}.

[figure]

Main features of these coefficients

1. 1 ≤ J(X) ≤ 2.

2. J(ℓ_{2}^{n}) = \sqrt{\frac{2n}{n+1}}.

3. If dim(X)=n, J(X) = 1 if and only if X = ℓ_{∞}^{n}.

4. J(ℓ_{2}) = \sqrt{2}.

5. ([A-X 93])

J(E_{β}) = \{ \begin{matrix}β\sqrt{2} & 1 ≤ β ≤ \sqrt{2} \\ 2 & \sqrt{2} < β < ∞ \end{matrix}

6. J(X) = 1 if and only if X = C(T) for a stonian T.

7. If the compact Hausdorff space T is not extremely disconnected, then for every finitecodimensional subspace E of C(T) we have J(E) = 2.

8. J(X) ≤ J_{s}(X).

9. If (X, ∥ · ∥) is a dual space, J(X) = sup{2r_{X}(K) : K finite, K ⊂ X, diam(K) = 1}.

10. If (X, ∥ · ∥) is a reflexive Banach space, then J_{s}(X) = sup{2r_{conv (K)}(K) : K finite, K ⊂ X, diam(K) = 1}.

11. If (X, ∥ · ∥) is a nonreflexive Banach space, then J_{s}(X) = 2.

12. For every infinite dimensional space (X, ∥ · ∥) , J_{s}(X) ≥ \sqrt{2}.

13. For every n-dimensional space (X, ∥ · ∥) , J_{s}(X) ≤ \frac{2n}{n+1}.

14. For every n and every ε > 0 we have

J_{s}(X) ≤ 2 max \{1 − \frac{1 − ε}{n!ε}, 1 − δ_{X}^{(n)}(ε)\} .

25 Bynum’s coefficient of (uniformly) normal structure

Notation: Let A be a nonempty bounded subset of X.

r(A) := inf_{a∈A} {sup{∥a − y∥ : y ∈ A}}

is called the Chebyshev self-radius of the set A.

Bynum in 1980 [By 80] defined the following coefficient of (uniform) normal structure:

N(X) := inf \{\frac{diam(C)}{r(C)} : C bounded, convex, C ⊂ X, diam(C) > 0\} .
[figure]

Geometric properties in terms of this constant Definition 13 A Banach space (X, ∥ · ∥) with N(X) > 1 is said to have uniform normal

structure.

[figure]

Separation of this property The Banach space (X, ∥ · ∥) obtained by the ℓ_{2} direct sum of the sequence of Banach spaces

(ℓ_{n}) is (UCED), and hence it has normal structure. Nevertheless N(X) = 1.

[figure]

Main features of this coefficient

1. ([B 84], [M 84]) If N(X) > 1 then X is reflexive.

2. ([CM 85]) If C is a nonempty weakly compact convex subset of X and T : C → C is uniformly k-lipschitzian with k < \sqrt{N(X)} then T has a fixed point in C.

3. Bynum’s lower bound of N(X) ([By 80]). If X is a reflexive Banach space,

N(X) ≥ \frac{1}{1 − δ_{X}(1)}.

4. Amir’s lower bound of N(X) ([Am 85])

N(X) ≥ sup \{(max \{1 − \frac{1}{n!n}(1 − ε), 1 − δ^{(k)}(ε)\})^{−1} : 0 ≤ ε ≤ 1\} .

5. ([Kh 87]) If lim inf_{t→0} \frac{ρ_{X}(t)}{t} < \frac{1}{2} then N(X) > 1.

6. Prus’ lower bound for N(X) ([Pr 96]).

N(X) ≥ (inf \{1 + ρ_{X}(t) − \frac{t}{2} : t ∈ [0, \frac{1}{2}]\})^{−1} .

7. ([By 80]) For a Hilbert space H, N(H) = \sqrt{2}.

8. ([CM 85]) For 1 ≤ β < \sqrt{2}, N(E_{β}) = \frac{\sqrt{2}}{β}. (For β ≥ \sqrt{2}, N(E_{β}) = 1).

9. ([By 80]) For X, Y isomorphic Banach spaces, N(X) ≤ d(X, Y)N(Y), where d(X, Y) is the Banach -Mazur distance between the Banach spaces X, Y.

10. ([Pr 90], [Do 91]) For 1 < p < ∞,

N(ℓ_{p}) = N(L^{p}([0, 1]) = min\{2^{\frac{1}{p}} , 2^{1− \frac{1}{p}} \}.

11. ([M 84]) If dim(X) = ∞ then N(X) ≤ \sqrt{2}.

12. W.L. Bynum defined in [By 80] the coefficient BS(X) as the supremum of the set of all numbers M with the property that for each bounded sequence (x_{n}) with asymptotic diameter A, there is some y in the closed convex hull of the (range of the ) sequence such that M lim sup_{n} ∥x_{n} − y∥ ≤ A.

Here

diam_{a} ((x_{n})) := lim_{n→∞} (sup ∥x_{i} − x_{j}∥ : i, j ≥ n)

is called the asymptotic diameter of the sequence (x_{n}), and

r_{a} ((x_{n})) := inf \{lim sup ∥x_{n} − x∥ : x ∈ \overline{conv} \{x_{n} : n = 1, … \}\}

is called the asymptotic radius of the sequence (x_{n}).

An equivalent definition is

BS(X) := inf \{\frac{diam_{a}((x_{n}))}{r_{a}((x_{n}))} : (x_{n}) is a nonconvergent bounded sequence in X\} .

It was shown in [Lm 83] that for all Banach space X, N(X) = BS(X).

13. ([CM 85] Suppose that, C is a weakly compact convex subset of X and T : C → C is k-uniformly Lipschitzian on C with k < \sqrt{N(X)}, then T has a fixed point.

26 Bynum’s weakly convergent sequence coefficient

It was defined by L.B. Bynum in [By 80] as follows

W CS(X) is the supremum of the set of all numbers M with the property that for each weakly convergent sequence (x_{n}) with asymptotic diameter A, there is some y in the closed convex hull of the (range of the) sequence such that M lim sup_{n} ∥x_{n} − y∥ ≤ A.

This is probably one of the Banach space constants most widely studied, although with considerable confusion because there are many equivalent definitions. We will follow in this summary the illuminating work by Sims and Smith [S-S 99]. See also the Ph.D. dissertation of M.A. Smyth [Smy 98].

An equivalent definition is (see [Lm 83])

W CS(X) := inf \{\frac{diam_{a}((x_{n}))}{r_{a}((x_{n}))} : (x_{n}) is a weakly (not strongly) convergent sequence in X\} .

[figure]

Geometrical properties in terms of this constant

1. Definition 14 We quote [S-S 99], p. 500.

Some authors have said that a space X has weak uniform normal structure if W CS(X) > 1. We shall say that X satisfies Bynum’s condition if this inequality holds.

2. Recall that a Banach space has the generalized Gossez-Lami Dozo property (GGLD) if for

every weakly null sequence (x_{n}) such that lim ∥x_{n}∥ = 1 we have that D[(x_{n})] > 1, where

D[(x_{n})] := lim sup_{m} (lim sup_{n} ∥x_{m} − x_{n}∥) .

This property was defined by A. Jiménez-Melado in [J 92]. In the same work there was defined the following coefficient in order to obtain stability results for (GGLD) in terms of the Banach-Mazur distance:

β(X) := inf{D[(x_{n})] : x_{n} ⇀^{w} 0, ∥x_{n}∥ → 1}.

Obviously (X, ∥·∥) has property (GGLD) if β(X) > 1. Moreover (GGLD) implies (WNS).

[figure]

Separation of this property

Example 8 The space c_{0} equivalently renormed by

∥(x_{n})∥ := ∥(x_{n})∥_{∞} + ∑_{n} \frac{|x_{n}|}{2^{n}},

was considered by A. Jiménez-Melado [J 92]. It enjoys Opial condition, and hence (WNS), but lacks (GGLD).

[figure]

Main features of this coefficient

1. ([S-S 99])

W CS(X) = inf \{\frac{diam_{a}((x_{n}))}{r_{a}((x_{n}))} : x_{n} ⇀^{w} 0, x_{n} ↛ 0\} .

’diam_{a}’ can be replaced with ’diam’ in the above equality.

2. ([S-S 99] and the references therein)

The following constant are equal.

(1) W CS(X).

(2) inf{diam_{a}((x_{n})) : x_{n} ⇀^{w} 0, ∥x_{n}∥ → 1}.

(3) β(X).

(4) inf{γ((x_{n})) : x_{n} ⇀^{w} 0, ∥x_{n}∥ → 1}.

(5) inf{α((x_{n})) : x_{n} ⇀^{w} 0, ∥x_{n}∥ → 1}.

3. ([By 80]) For a reflexive Banach space (X, ∥ · ∥) , 1 ≤ N(X) ≤ BS(X) ≤ W CS(X) ≤ 2.

4. ([A-D 93]) If X is a reflexive infinite dimensional space such that the duality mapping is

continuous, then

1 ≤ N(X) ≤ \frac{2}{J(X)} ≤ W CS(X) ≤ 2.

5. ([D-L 92])

W CS(X) ≥ \frac{1}{∏_{n=0}^{∞}(1 − δ_{X}(2^{−n}))}.

6. ([By 80]) Let X, Y be isomorphic Banach spaces. Then,

W CS(X) ≤ d(X, Y)W CS(Y).

(See also [A-D-L 97], p. 119)

7. ([J 92]) If W CS(X) = β(X) > 1 and d(X, Y) < β(X) then Y has property GGLD.

8. ([By 80]) For p ≥ 1, W CS(ℓ_{p}) = 2^{\frac{1}{p}}.

9. For a Hilbert space H, W CS(H) = \sqrt{2}.

10. ([A-X 93])

W CS(E_{β}) = \{ \begin{matrix}\frac{\sqrt{2}}{β} & 1 ≤ β\sqrt{2} \\ 1 & \sqrt{2} < β < ∞. \end{matrix}

11. ([A-D-L 97]) W CS(c_{0}) = 1.

12. For any Banach space we have

W CS(X) ≥ 1 + r_{X}(1).

(See [LTX 95]). This inequality may be strict, as in the following example. Example 9 ([Kc 99]) Let X = ℓ_{2} ⊕ ℝ equipped with the norm

∥(x, r)∥ := ∥x∥_{2} + max \{|r| − \frac{1}{2}∥x∥_{2}, 0\}

where ∥.∥_{2} denotes the euclidean ℓ_{2} norm. The space X has the nonstrict Opial property, r_{X}(1) = \frac{1}{4} and W CS(X) = \sqrt{2}.

13. ([D-L-X 96]) W CS(ℓ_{p,q}) = min\{2^{\frac{1}{p}} , 2^{\frac{1}{q}} \}.

14. ([D-L 92]) If (X, ∥ · ∥) is a non-Schur Banach space

W CS(X) ≥ lim_{ε→1^{−}} \frac{1}{1 − Δ_{X,β}(ε)}.

15. ([Do 95]) Let X be a reflexive Banach space. Then

sup_{0≤ε≤a} \frac{tεW CS(X^{∗})}{4} − Δ_{X,σ}(ε) ≤ Γ_{X^{∗}}(t) ≤ sup_{0≤ε≤a} \frac{tε}{W CS(X)} − Δ_{X,σ}(ε)

for every t > 0, where a = σ(B_{X}). If, in addition, X^{∗} satisfies the Opial condition then

sup_{0≤ε≤a} \frac{tε}{2} − Δ_{X,σ}(ε) ≤ Γ_{X^{∗}}(t).

16. ([Do 95]) Let X be a reflexive Banach space. Then

W CS(X) lim_{t→0} \frac{Γ_{X^{∗}}(t)}{t} ≤ ε_{0,σ}(X) ≤ \frac{4}{W CS(X^{∗})} lim_{t→0} \frac{Γ_{X^{∗}}(t)}{t}.

17. ([Do 95]) Let X be a reflexive Banach space satisfying the Opial condition. Then

sup_{0≤ε≤a} \frac{tεW CS(X)W CS(X^{∗})}{4} − Δ_{X,β}(ε) ≤ Γ_{X^{∗}}(t) ≤ sup_{0≤ε≤a} tε − Δ_{X,β}(ε)

for every t > 0, where a = σ(B_{X}). If, in addition, X^{∗} satisfies the Opial condition then

sup_{0≤ε≤a} \frac{tεW CS(X)}{2} − Δ_{X,β}(ε) ≤ Γ_{X^{∗}}(t).

18. ([Do 92]) Let (X_{n}, |.|_{n}) be a sequence of reflexive Banach spaces. Let 1 < p < ∞. Let X be the ℓ_{p}-direct sum of (X_{n}, |.|_{n}). Then,

W CS(X) = inf\{W CS(X_{i}), 2^{\frac{1}{p}} : i ∈ ℕ\}.

19. ([Do 92]) Let (X_{1}, |.|_{1}), …, (X_{n}, |.|_{n}) be a finite sequence of reflexive Banach spaces. Let Z be ℝ^{n} with a monotone norm. Then,

W CS ((X_{1} ⊕ … ⊕ X_{n})_{Z}) = min{W CS(X_{i}) : i = 1, …, n}.

20. ([Pr 90], [Do 91]) Let (Ω, Σ, μ) be a σ-finite measure space. Let 1 ≤ p < +∞ and assume that L^{p}(Ω) is infinite dimensional. Then

W CS(L^{p}(Ω)) = N(L^{p}(Ω)) = min\{2^{\frac{1}{p}} , 2^{1− \frac{1}{p}} \}

if either p ≥ 2 or μ is not purely atomic.

21. ([D-X 95], [A-D-L 97]) Suppose that W CS(X) > 1, C is a weakly compact convex subset of X and T : C → C is asymptotically regular on C. If

lim inf |T^{n}| < \sqrt{W CS(X)},

then T has a fixed point.

27 Gao-Lau coefficient

This was defined in an attempt to simplify Shäfer’s notion of girth and perimeter of the unit ball of a normed space [Sc 76].

G(X) := sup {∥x + y∥ ∧ ∥x − y∥ : x, y ∈ S_{X}} .
[figure]

Geometrical properties in terms of this constant

1. ([G-L 90]) G(X) < 2 if and only if X is uniformly nonsquare.

2. ([G-L 91]) If G(X) < \frac{3}{2} then (X, ∥ · ∥) has uniform normal structure.

[figure]

Main features of this coefficient

1. ([G-L 90]) For any Banach space (X, ∥ · ∥) , \sqrt{2} ≤ G(X) ≤ 2.

2. ([G-L 90]) G(ℓ_{p}) = G(L^{p}) = max\{2^{\frac{1}{p}} , 2^{1− \frac{1}{p}} \}.

3. ([G-L 91]) G(X) < ε if and only if d_{X}(ε) > 1 − \frac{ε}{2}.

4. ([Pr 91]) N(X) ≤ G(X) + 1 − ((G(X) + 1)^{2} − 4)^{\frac{1}{2}} .

5. ([G-L 91]) For any isomorphism T from X to ℓ_{p} or L^{p}, 1 < p < ∞,

G(X) ≤ ∥T ∥∥T^{−1}∥ max \{2^{\frac{1}{p}} , 2^{1− \frac{1}{p}} \} .

6. ([G-L 91]) For any Banach space (X, ∥ · ∥) and for any nontrivial ultrafilter U on ℕ,

G(X_{U}) = G(X).

7. ([G-L 91])

G(X) = sup \{ε ≥ 0 : δ_{X}(ε) ≤ 1 − \frac{1}{2}ε\} .

8. ([G-L 90]) If X and Y are Banach spaces and T : X → Y is a (bicontinuous) isomorphism,

then

(∥T ∥∥T^{−1}∥)^{−1} ≤ \frac{G(X) + 2}{G(Y) + 2} ≤ ∥T ∥∥T^{−1}∥.

9. ([G-L 91]) If G(Y) < \frac{3}{2} and

Δ(X, Y) < ln (\frac{7}{2G(Y) + 2}) ,

then X has uniform normal structure.

Here Δ(X, Y) := inf {ln(∥T ∥∥T^{−1}∥) : T ∈ I(X, Y)}.

28 A measure of the degree of WORTHwileness, [Si 94]

w(X) := sup{λ : λ lim inf ∥x_{n} + x∥ ≤ lim inf ∥x_{n} − x∥ : x_{n} ⇀^{w} 0 x ∈ X}.

[figure]

Geometrical properties in terms of this coefficient

1. The Banach space (X, ∥ · ∥) has property (k) if

w(X) > max \{\frac{1}{2}ε, 1 − δ_{X}(ε)\}

for some positive ε.

Recall that a Banach space (X, ∥ · ∥) has property (k) if there exists k ∈ [0, 1) such that whenever x_{n} ⇀^{w} 0, ∥x_{n}∥ → 1 and lim inf_{n} ∥x_{n} − x∥ ≤ 1 we have ∥x∥ ≤ k. One has that property (k)⇒ (W NS). Property (k) is equivalent to r(1) > 0. (See [D 01])

[figure]

Main features of this coefficient

1. k(L_{p}([0, 1])) = (1 − 2^{−p})^{\frac{1}{p}}.

2. (See [J-L 96]) A Banach space (X, ∥ · ∥) has the (FPP) whenever

\frac{ε_{0}(X)}{4} + \frac{1}{2w(X)} < 1.

29 A coefficient related with (NUS) property

It was defined by J. García-Falset in [GF 94]

R(X) := sup{lim inf_{n→∞} ∥x_{n} + x∥ : x, x_{n} ∈ B_{X}(n = 1, …), x_{n} ⇀^{w} 0, }.

[figure]

Geometrical properties in terms of this coefficient

1. Let (X, ∥ · ∥) a Banach space. The following conditions are equivalent:

(a) X is (WNUS).

(b) X is reflexive and R(X) < 2.

Recall that (X, ∥ · ∥) is (WNUS) if for some ε > 0 there exists μ > 0 such that if 0 < t < μ and (x_{n}) is a basic sequence in B_{X} then there exists k > 1 so that ∥x_{1} + x_{n}∥ ≤ 1 + εt.

[figure: Main features of this coefficient](See [GF 94] and [GF 98]).

1. 1 ≤ R(X) ≤ 2.

2. If X is finite dimensional, R(X) = 1.

3. Let X be a weakly orthogonal Banach Lattice. Then R(X) ≤ α(X) where α(X) is the Riesz angle of X.

4. R (c_{0}) = R (ℓ_{1}) = 1.

5. For 1 < p < ∞, R (ℓ_{p})) = 2^{\frac{1}{p}}.

6. R(c) = 2.

7. Let (X, ∥ · ∥) be a Banach space with the property (WORTH). Then R(X) ≤ G(X).

8. R(ℓ_{p,∞}) ≤ 2^{\frac{1}{p}}.

9. ([D 01]) If R(X) = 1 then the following are equivalent. (X, ∥ · ∥) is a Schur space; (X, ∥ · ∥) has (WNS); (X, ∥ · ∥) has the (KK) property; c_{0} ↛ X. Recall that a Banach space is said to have the (KK) property whenever ∥x_{n} − x∥ → 0 if

x_{n} ⇀^{w} x and ∥x_{n}∥ → ∥x∥.

10. If R(X) < 2 then X has the weak Banach-Saks property.

11. ε_{0}(X) < 1 ⇒ R(X) < 2.

12. R(X) < 2 ⇒ (X, ∥.∥) has (WFPP).

13. A Banach space (X, ∥ · ∥) has the (WFPP) if there exists a isomorphic Banach space Y such that d(X, Y)R(Y) < 2.

14. If \frac{ε_{0}(X)}{w(X)} < 2 then R(X) < 2.

30 Domínguez generalization, [Do 96]

Given a nonnegative real number a, one defines

R(a, X) := sup{lim inf ∥x + x_{n}∥},

where the supremum is taken over all x ∈ X with ∥x∥ ≤ a and all weakly null sequences (x_{n}) in the unit ball of X such that

D(x_{n}) = lim sup_{n} (lim sup_{m} ∥x_{n} − x_{m}∥) ≤ 1 .

Moreover we put

M(X) := sup \{\frac{1 + a}{R(a, X)} : a ≥ 0\} .

[figure: Main features of this coefficient](See [Do 96])

1. If R(a, X) < 1 + a then (X, ∥ · ∥) has the (WFPP).

2. Let X, Y be isomorphic Banach spaces. Then

R(a, Y) ≤ d(X, Y)R(a, X)

for every nonnegative number a.

3. If M(X) > 1, then (X, ∥ · ∥) has the (WFPP).

4. If M(X) > 1 and Y is a Banach space which is isomorphic to X and d(X, Y) < M(X), then Y has the (WFPP).

5. R(0, X) = \frac{1}{W CS(X)}.

6. M(X) ≥ W CS(X).

7. M(ℓ_{2,∞}) = \sqrt{2} > W CS(ℓ_{2,∞} = 1.

8. Let X be a reflexive Banach space and denote

Γ(X) := inf \{1 + Γ(s) − \frac{s}{2} : s ∈ [0, 1]\} .

Then

(a) For every a ≥ 0, R(a, X) ≤ 1 + aΓ(X).

(b) M(X) ≥ \frac{1}{Γ(X)}.

9. Let X be a reflexive Banach space and denote

Γ′(X) := inf \{1 + Γ(s) − \frac{sW CS(X)}{2} : s ∈ [0, 1]\} .

Then

(a) M(X) ≥ \frac{1}{Γ′(X)}.

(b) M(X) > 1 if Γ′(0) < \frac{W CS(X)}{2}.

(c) M(X) > 1 if Γ′(0) < \frac{1}{2}.

10. ([A-D-L 97]) Let X be a Banach space and denote

β_{X} := inf \{1 + β_{X}^{k}(s) − \frac{s}{2k} : s ∈ [0, 1]\} .

Then

M(X) ≥ \frac{1 + 2k}{1 + 2kβ_{X}} .

11. Let X be a reflexive Banach space. If c ∈ (0, 1) satisfies r_{X^{∗}}(c) > 0 then

R(a, X) ≤ max \{1 + ac, a + \frac{1}{1 + r_{X^{∗}}(c)}\} .

In particular, if r_{X^{∗}}(1) > 0 then R(a, X) < 1 + a and M(X) > 1.

12. For 1 < p < ∞,

R(a, ℓ_{p}) = (a^{p} + \frac{1}{2})^{\frac{1}{p}} ⇒ M(ℓ_{p}) = (1 + 2^{\frac{1}{p−1}} )^{\frac{p−1}{p}} .

13.

R(a, ℓ_{p,∞}) = (1 + a^{p})^{\frac{1}{p}} ⇒ M(ℓ_{p,∞}) = 2^{\frac{p−1}{p}} .

14. For a Hilbert space X, M(X) = \sqrt{3}.

15. ([D-J 98])

M(E_{β}) = \{ \begin{matrix}\sqrt{3} & β ≤ \sqrt{\frac{3}{2}} \\ \frac{\sqrt{2}}{β} (1 + \sqrt{\frac{β^{2}−1}{2}}) & \sqrt{\frac{3}{2}} < β < \sqrt{2} \\ 1 + \frac{1}{\sqrt{2}} & \sqrt{2} ≤ β. \end{matrix}

16. ([D-J 98]) For 1 ≤ p < ∞ and 1 ≤ q < ∞,

M(ℓ_{p,q}) = \{ \begin{matrix}2^{\frac{1}{q}} [1 + (\frac{1}{2})^{\frac{p}{q(p−1)}} ]^{\frac{p−1}{p}} & p ≤ q \\ 2^{\frac{1}{p}} [1 + (\frac{1}{2})^{\frac{p}{q(p−1)}} ]^{\frac{p−1}{p}} & q < p. \end{matrix}

17. R(a, c_{0}) = a ⇒ M(c_{0}) = 2.

18. (See [J-L 00]) If we define, for B ≥ 1 the number C_{X}(B) := {c ≥ 0 : r_{X}(c) ≤ B − 1} and

| · | is an equivalent renorming of the Banach space (X, ∥ · ∥) such that for all x ∈ X ∥x∥ ≤ |x| ≤ B∥x∥,

and

B < sup \{\frac{1 + a}{R (\frac{a}{B} C_{X}(B), X)} : a ≥ 0\} ,

then (X, | · |) has the (WFPP).

31 Lifschitz and Dominguez-Xu coefficients for uniformly lipschitzian mappings

In 1975, Lifschitz ([Lf 75]) introduced the coefficient κ(M) for a metric space (M, d) as follows

κ(M) := sup{β > 0 : ∃α > 1 s.t. ∀x, y ∈ M, ∀r > 0, d(x, y) > r
B[x, βr] ∩ B[y, αr] ⊂ B[z, r] for some z ∈ M}.

He proved that if (M, d) is a complete bounded metric space and T is a k-uniformly Lipschitzian selfmapping of M with k < κ(M), then T has a fixed point. For a Banach space (X, ∥ · ∥) , it is often denoted by κ_{0}(X) := inf{κ(M)} where the infimum is taken over all the closed convex and bounded subsets of X.

The following definitions are inspired on this Lifschitz’s coefficient, and were given in [D-X 95] by T. Domínguez-Benavides and H.K. Xu.

Let M be a bounded convex subset of X. A number b ≥ 0 is said to have property (P) with respect to M if there exists some a > 1 such that for all x, y ∈ M and r > 0 with ∥x − y∥ ≥ r and each weakly convergent sequence (x_{n}) in M for which

lim sup ∥x_{n} − x∥ ≤ ar
lim sup ∥x_{n} − y∥ ≤ br

there exists some z ∈ M such that lim inf ∥x_{n} − z∥ ≤ r.

Then one defines

κ̃(M) := sup{b > 0 : b has property (P) w.r.t. M}.

κ̃(X) := inf{κ̃(M) : M as above}.

[figure]

Main features of these coefficients

1. ([A-D-L 97], p. 145) Let (X, ∥ · ∥) a Banach space and h a solution of the equation

h (1 − δ_{X} (\frac{1}{h})) = 1.

Then h ≤ κ_{0}(X). Furthermore ε_{0}(X) < 1 if and only if κ_{0}(X) > 1.

2. ([A-X 93])

κ_{0}(L^{p}) ≥ \{ \begin{matrix}[1 + \frac{1+τ_{p}^{p−1}}{(1+τ_{p})^{p−1}} ]^{\frac{1}{p}} & p > 2 \\ \sqrt{p} & p ≤ 2 \end{matrix}

where τ_{p} ∈ (0, 1) is the unique solution of the equation

(p − 2)τ^{p−1} + (p − 1)τ^{p−2} = 1.

3. ([A-D-L 97], p. 145, [A-X 93])

1 ≤ κ_{0}(X) ≤ N(X) ≤ W CS(X) ≤ 2
1 ≤ κ_{0}(X) ≤ N(X) ≤ \sqrt{2}
1 ≤ κ_{0}(X) ≤ N(X) ≤ \frac{2}{J(X)} ≤ 2
κ(X) ≤ \frac{2}{J(X)}.

4. ([Lf 75]) κ_{0}(X) ≥ \frac{1}{1−δ_{X}(1)}.

5. ([Lf 75]) κ_{0}(H) = \sqrt{2} whenever H is an infinite dimensional Hilbert space.

6. In [D-X 95] one can see easier equivalent definitions of κ̃(X) when (X, ∥ · ∥) is a Banach space satisfying the uniform Opial condition.

7. ([D-X 95]) If (X, ∥ · ∥) is a Banach space satisfying the uniform Opial condition then κ̃(X) = 1 + r_{X}(1).

8. ([D-X 95]) For 1 < p < ∞, κ̃(ℓ_{p}) = 2^{\frac{1}{p}}.

9. ([D-X 95]) Let (X, ∥ · ∥) a Banach space with the uniform Opial condition and M be a bounded convex subset of X. Then κ̃(M) ≥ h where h is the unique solution of the equation

t [1 − Δ_{X} (\frac{1}{t})] = 1.

10. ([D-X 95]) Let (X, ∥ · ∥) a Banach space with the uniform Opial condition and M be a bounded convex subset of X. Then κ̃(M) ≥ h_{0} where

h_{0} := sup \{t > 1 : Δ′_{X} (\frac{W CS(X)}{t}) + \frac{1}{t} > 1\} .

11. ([D-X 95], [A-D-L 97]) If (X, ∥ · ∥) is a Banach space satisfying the Opial condition, or X

is reflexive then

κ̃(X) ≤ W CS(X).

Example 10 Let X be ℓ_{2} renormed by

∥x∥ := [|x_{1}|^{p} + (∑_{n=2}^{∞} |x_{n}|^{2})^{\frac{p}{2}} ]^{\frac{1}{p}}

where 2 < p < ∞.

κ̃(X) ≤ 2^{\frac{1}{p}} < W CS(X) = \sqrt{2}.

12. ([A-D-L 97], Theorem IX.2.3) If (X, ∥ · ∥) is a Banach space satisfying the uniform Opial condition, then

κ̃(X) ≥ \frac{1}{1 − Δ_{X,β}(1^{−})}.

13. ([D-X 95]) Suppose that C is a weakly compact convex subset of the Banach space (X, ∥·∥) and T : C → C is a uniformly Lipschitzian mapping. If

lim inf_{n} |T^{n}| < κ̃(C),

then T has a fixed point in C. (Here |T^{n}| denotes the exact Lipschitz constant of T^{n}).

14. ([Do 98]) Let (X, ∥ · ∥) a Banach space, C a closed convex bounded subset of X and T : C → C a k-uniformly lipschitzian mapping. If

k < \frac{1 + \sqrt{1 + 4N(X)(κ_{0}(X) − 1)}}{2}

then T has a fixed point.

15. ([Do 98])

κ_{0}(X) ≤ \frac{1 + \sqrt{1 + 4N(X)(κ_{0}(X) − 1)}}{2} ≤ N(X).

16. ([Zh 92]) If β ≥ \frac{1}{2} \sqrt{5}, then κ_{0}(E_{β}) = 1.

17. ([Do 98]) If 1 ≤ β ≤ \frac{1}{2} \sqrt{5}, then

κ_{0}(E_{β}) = (1 + \frac{1}{λ^{2}} − \frac{2}{λ^{2}} \sqrt{λ^{2} − 1})^{\frac{1}{2}} .

18. ([Do 98]) Let (X, ∥ · ∥) be a reflexive Banach space, C a bounded closed convex subset of X and T : C → C an asymptotically regular mapping. If

lim inf |T^{n}| < \frac{1 + \sqrt{1 + 4W CS(X)(κ̃(X) − 1)}}{2}

then T has a fixed point.

32 Coefficients for asymptotic normal structure, [B-K-R 98]

They were defined in [B-K-R 98] in order to get a quantitative description of the asymptotic normal structure, playing a similar role to that Bynum’s coefficients for normal structure.

For a subsequence (x_{n_{i}}) of a bounded sequence (x_{n}) in X, we will denote by r_{a} ((x_{n_{i}})) the asymptotic radius of this subsequence with respect to the set \overline{co} (\{x_{n} : n ∈ ℕ\}), i.e.

r_{a} ((x_{n_{i}})) = inf \{lim sup_{i} ∥x − x_{n_{i}}∥ : x ∈ \overline{co} (\{x_{n} : n ∈ ℕ\})\} .

Moreover the bounded sequence (x_{n}) in X is asymptotically regular (a.r.) whenever x_{n}−x_{n+1} → 0. Now we define

AN(X) := sup{k : k. inf_{(x_{n_{i}})} r_{a} ((x_{n_{i}})) ≤ diam_{a}((x_{n}))

for each a.r. bounded sequence (x_{n})} .

If we add in this definition the condition that the sequence (x_{n}) is such that the set \overline{co} (\{x_{n} : n ∈ ℕ}) is weakly compact, then we get the asymptotic normal structure coefficient with respect to the weak topology

w − AN(X) := sup{k : k. inf_{(x_{n_{i}})} r_{a} ((x_{n_{i}})) ≤ diam_{a}((x_{n}))

for each a.r. bounded sequence (x_{n})

with \overline{co} (\{x_{n} : n ∈ ℕ\}) weakly compact\} .

[figure]

Geometrical properties in terms of these constants

1. Definition 15 If AN(X) > 1 we say that (X, ∥·∥) has uniform asymptotic normal structure, (UAN).

2. Definition 16 If w − AN(X) > 1 we say that (X, ∥ · ∥) has uniform asymptotic normal structure with respect to the weak topology, (w-UAN).

Main features of these coefficients 1. 1 ≤ AN(X) ≤ w − AN(X).

[figure]

2. 1 ≤ W CS(X) ≤ w − AN(X).

3. In the definitions of w − AN(X) we can replace diam_{a}((x_{n})) by diam((x_{n})).

4. If a Banach space (X, ∥ · ∥) has AN(X) > 1 then it is reflexive.

5. w − AN(X) > 1 ⇒ (W F P P).

6. Let (X, ∥ · ∥) and (Y, | · |) be isomorphic Banach spaces. Then we have

\begin{matrix}AN(X) ≤ d(X, Y)AN(Y), \\ w − AN(X) ≤ d(X, Y).w − AN(Y). \end{matrix}

7. AN(X) = ∞ if and only if (X, ∥ · ∥) is finite dimensional.

8. w − AN(X) = ∞ if and only if (X, ∥ · ∥) is a Schur space.

9. AN(ℓ_{p}) = 2^{\frac{1}{p}}.

10. Example 11 For β, p > 1 let us consider ℓ_{p} endowed with the norm

∥x∥_{β} := max \{∥x∥_{∞}, \frac{1}{β} ∥x∥_{p}\} .

We denote this space by X_{β}^{p}. If we take the space Z := X_{√\overline{2}}^{2}×ℓ_{1} equipped with the ℓ_{1}-norm then this space is nonreflexive. Thus, AN(Z) = 1. But w − AN(Z) = \sqrt{2}.

33 Coefficient for Semi-Opial property, [B-K-R 98]

It was defined in [B-K-R 98]. Recall that a Banach space is said to have the semi-Opial (weak semi-Opial) property, (SO) ((w-SO)) for short, if for each bounded nonconstant asymptotically regular sequence (x_{n}) in X (with weakly compact closed convex hull), there exists a subsequence (x_{n_{i}}) weakly convergent to x, such that

lim inf_{i} ∥x − x_{n_{i}}∥ < diam({x_{n}}).

Now we define the semi-Opial coefficient with respect to the weak topology as follows

w − SOC(X) := sup{k : k. inf_{(x_{n_{i}}),x_{n_{i}} ⇀^{w} y} r_{a} (y, (x_{n_{i}})) ≤ diam_{a}((x_{n}))

for each a.r. bounded sequence (x_{n})

with \overline{co} (\{x_{n} : n ∈ ℕ\}) weakly compact\} .

[figure]

Geometrical properties in terms of this constant

1. Definition 17 If w − SOC(X) > 1 we say that (X, ∥ · ∥) has uniform semi-Opial property,with respect to the weak topology, (w-USO).

[figure]

Main features of this coefficient

1. 1 ≤ W CS(X) ≤ w − SOC(X) ≤ w − AN(X).

2. In the definitions of w − SOC(X) we can replace diam_{a}((x_{n})) by diam((x_{n})).

3. If a Banach space (X, ∥ · ∥) has the nonstrict Opial property, then w − SOC(X) = w − AN(X).

4. The space L^{p}([0, 1]), 1 < p < ∞, p ≠ 2, has w − USO property but does not satisfy the nonstrict Opial condition.

5. Let (X, ∥ · ∥) and (Y, | · |) be isomorphic Banach spaces. Then we have w − SOC(X) ≤ d(X, Y).w − SOC(Y).

6. w − SOC(X) = ∞ if and only if (X, ∥ · ∥) is a Schur space.

7. Suppose that X = W ⊕ Z where W is a closed subspace of X, Z is a Schur space, and the projection onto W has norm 1. Then we have w − SOC(X) = w − SOC(W).

8. Let (X, ∥ · ∥) and (Y, | · |) be Banach spaces. If (X, ∥ · ∥) is w − USO and W CS(Y) > 1 then (X × Y)_{ℓ_{p}} (1 ≤ p < ∞), is also w − USO.

9. w − SOC(ℓ_{p}) = 2^{\frac{1}{p}}. 10.

w − SOC(X_{β}^{p}) = max [1, min (2^{\frac{1}{p}} , \frac{4^{\frac{1}{p}}}{β} )] .

11. If we take the space Z := X_{√\overline{2}}^{2} × ℓ_{1} equipped with the ℓ_{1}-norm then this space is nonreflexive. Thus, AN(Z) = 1. But w − SOC(Z) = \sqrt{2}.

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