Preserved and unpreserved extreme points A. J. Guirao 1 , V. - - PowerPoint PPT Presentation

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Preserved and unpreserved extreme points

• A. J. Guirao1,
• V. Montesinos1,
• V. Zizler2

1Instituto de Matemática Pura y Aplicada, Universidad Politécnica de Valencia,

Spain 2Alberta University, Edmonton, Alberta, Canada

Aleksander Pełczy´ nski Memorial Conference

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Alexander Pełczy´ nski

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Preserved and unpreserved extreme points

• A. J. Guirao1,
• V. Montesinos1,
• V. Zizler2

1Instituto de Matemática Pura y Aplicada, Universidad Politécnica de Valencia,

Spain 2Alberta University, Edmonton, Alberta, Canada

Aleksander Pełczy´ nski Memorial Conference

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Outline

1

Basic facta

2

Earlier and related results

3

Some other definitions and auxiliary results

4

The main results

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Extreme points

X Banach. BX closed unit ball, SX unit sphere

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Extreme points

X Banach. BX closed unit ball, SX unit sphere x ∈ SX is extreme if x = y + z 2 and y, z ∈ BX ⇒ y = z

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Extreme points

X Banach. BX closed unit ball, SX unit sphere x ∈ SX is extreme if x = y + z 2 and y, z ∈ BX ⇒ y = z

extreme nonextreme BX

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Extreme points

X Banach. BX closed unit ball, SX unit sphere x ∈ SX is extreme if x = y + z 2 and y, z ∈ BX ⇒ y = z

extreme nonextreme BX

BX X X∗∗ BX∗∗

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Remark

If X nonreflexive then BX ∗∗ has extreme points non in X.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Remark

If X nonreflexive then BX ∗∗ has extreme points non in X. BX X X∗∗ BX∗∗

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Remark

If X nonreflexive then BX ∗∗ has extreme points non in X. BX X X∗∗ BX∗∗ Reason: James’ Theorem

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

preserved extreme points

x ∈ SX is preserved extreme whenever extreme of BX ∗∗

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

preserved extreme points

x ∈ SX is preserved extreme whenever extreme of BX ∗∗

X

BX∗∗

all x ∈ SX preserved

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

preserved extreme points

x ∈ SX is preserved extreme whenever extreme of BX ∗∗

X

BX∗∗

all x ∈ SX preserved

Otherwise, unpreserved

X

BX∗∗

all x ∈ SX unpreserved

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

All extreme points are preserved in C(K), Lp, 1 ≤ p ≤ ∞

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

All extreme points are preserved in C(K), Lp, 1 ≤ p ≤ ∞

X BX∗∗

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

All extreme points are preserved in C(K), Lp, 1 ≤ p ≤ ∞

X BX∗∗

Question (Phelps’61) ∃ unpreserved?

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

All extreme points are preserved in C(K), Lp, 1 ≤ p ≤ ∞

X BX∗∗

Question (Phelps’61) ∃ unpreserved? Answer (Katznelson’61) Disk algebra

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Lemma (Rosenthal) e ∈ SX preserved extreme ⇔ {slices ∋ e} base of w-neighborhoods.

e BX

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

x0 xn

x0+xn 2

BX X BX∗∗ BX LUR all SX preserved extreme

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

x0 Strongly exposed

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

x0 Strongly exposed

x0 Denting

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

x0 Strongly exposed

x0 Denting

LUR ⇒ strongly exposed

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

x0 Strongly exposed

x0 Denting

LUR ⇒ strongly exposed ⇒ denting

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

x0 Strongly exposed

x0 Denting

LUR ⇒ strongly exposed ⇒ denting

x0 (w-) Strongly extreme

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

x0 Strongly exposed

x0 Denting

LUR ⇒ strongly exposed ⇒ denting

x0 (w-) Strongly extreme x0 Extreme

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

x0 Strongly exposed

x0 Denting

LUR ⇒ strongly exposed ⇒ denting

x0 (w-) Strongly extreme x0 Extreme

denting ⇒ strongly extreme

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

x0 Strongly exposed

x0 Denting

LUR ⇒ strongly exposed ⇒ denting

x0 (w-) Strongly extreme x0 Extreme

denting ⇒ strongly extreme⇒ w-strongly extreme

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

x0 Strongly exposed

x0 Denting

LUR ⇒ strongly exposed ⇒ denting

x0 (w-) Strongly extreme x0 Extreme

denting ⇒ strongly extreme⇒ w-strongly extreme (=preserved extreme [Godun–Lin– Troyanski’92])

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

x0 Strongly exposed

x0 Denting

LUR ⇒ strongly exposed ⇒ denting

x0 (w-) Strongly extreme x0 Extreme

denting ⇒ strongly extreme⇒ w-strongly extreme (=preserved extreme [Godun–Lin– Troyanski’92]) ⇒ extreme

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

x0 Strongly exposed

x0 Denting

LUR ⇒ strongly exposed ⇒ denting

x0 (w-) Strongly extreme x0 Extreme

denting ⇒ strongly extreme⇒ w-strongly extreme (=preserved extreme [Godun–Lin– Troyanski’92]) ⇒ extreme all SX strongly extreme ≡ MLUR (see Lajara, Pallarés, Troyanski’09)

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

x0 Strongly exposed

x0 Denting

LUR ⇒ strongly exposed ⇒ denting

x0 (w-) Strongly extreme x0 Extreme

denting ⇒ strongly extreme⇒ w-strongly extreme (=preserved extreme [Godun–Lin– Troyanski’92]) ⇒ extreme all SX strongly extreme ≡ MLUR (see Lajara, Pallarés, Troyanski’09) all SX extreme ≡ R

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Remark X RNP then BX has extreme points that are preserved.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Theorem TFAE (i) X fails RNP .

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Theorem TFAE (i) X fails RNP . (ii) [Schachermayer–Sersouri–Werner’89] ∀ ε ∃ | · | dist (ExtB∗∗

|·|, X) ≥ 1 − ε.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Theorem TFAE (i) X fails RNP . (ii) [Schachermayer–Sersouri–Werner’89] ∀ ε ∃ | · | dist (ExtB∗∗

|·|, X) ≥ 1 − ε.

(iii) [Bourgain’78, Stegall’63] ∃ | · | each extreme is unpreserved (and dist (ExtB∗∗

|·|, X) > 0).

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A renorming answer to Phelp’s question Theorem (Godun’85) TFAE

1

X nonreflexive.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A renorming answer to Phelp’s question Theorem (Godun’85) TFAE

1

X nonreflexive.

2

∃ | · |, some x ∈ S|·| extreme, and unpreserved.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A renorming answer to Phelp’s question Theorem (Godun’85) TFAE

1

X nonreflexive.

2

∃ | · |, some x ∈ S|·| extreme, and unpreserved.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A dramatic renorming answer to Phelps’s question. Theorem (Morris’83) X separable, c0 ֒ → X, then ∃ | · |, all points in SX unpreserved.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A dramatic renorming answer to Phelps’s question. Theorem (Morris’83) X separable, c0 ֒ → X, then ∃ | · |, all points in SX unpreserved. X

BX∗∗

all x ∈ SX unpreserved

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A dramatic renorming answer to Phelps’s question. Theorem (Morris’83) X separable, c0 ֒ → X, then ∃ | · |, all points in SX unpreserved. X

BX∗∗

all x ∈ SX unpreserved Separable problems: Since (c0 ֒ → X) ⇒ M ⇒ ¬RNP,

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A dramatic renorming answer to Phelps’s question. Theorem (Morris’83) X separable, c0 ֒ → X, then ∃ | · |, all points in SX unpreserved. X

BX∗∗

all x ∈ SX unpreserved Separable problems: Since (c0 ֒ → X) ⇒ M ⇒ ¬RNP, 1.- Characterize spaces with M

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A dramatic renorming answer to Phelps’s question. Theorem (Morris’83) X separable, c0 ֒ → X, then ∃ | · |, all points in SX unpreserved. X

BX∗∗

all x ∈ SX unpreserved Separable problems: Since (c0 ֒ → X) ⇒ M ⇒ ¬RNP, 1.- Characterize spaces with M 2.- Characterize spaces where each renorming has some extreme point preserved

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A little improvement

Theorem (Guirao, M., Zizler) X separable, c0 ֒ → X. Then ∃ | · |, all points unpreserved, one direction, uniform.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A little improvement

Theorem (Guirao, M., Zizler) X separable, c0 ֒ → X. Then ∃ | · |, all points unpreserved, one direction, uniform.

δ δ x0 BX∗∗ y∗∗

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Definition X polyhedral whenever BX ∩ F polyhedron ∀ finite-dimensional F.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Definition X polyhedral whenever BX ∩ F polyhedron ∀ finite-dimensional F.

BX F non-polyhedral

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Definition X polyhedral whenever BX ∩ F polyhedron ∀ finite-dimensional F.

BX F non-polyhedral BX F polyhedral

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Theorem (Fonf, Hájek’95) TFAE (i) X separable isomorphic to polyhedral.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Theorem (Fonf, Hájek’95) TFAE (i) X separable isomorphic to polyhedral. (ii) X has a countable James boundary.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Theorem (Fonf, Hájek’95) TFAE (i) X separable isomorphic to polyhedral. (ii) X has a countable James boundary. (iii) X separable and equivalent norm C∞-smooth away from the origin and depends locally on a finite number of coordinates.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Theorem (Fonf’80) X with countable James boundary ⇒ c0-saturated

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Theorem (Fonf’80) X with countable James boundary ⇒ c0-saturated Theorem (Rodé, Godefroy) X with a separable James boundary ⇒ X separable Asplund. (Note: their result is much more general)

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

The main results I

Theorem (Guirao, Zizler, M.) X with countable James boundary.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

The main results I

Theorem (Guirao, Zizler, M.) X with countable James boundary. Then ∃ | · | C∞-smooth,

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

The main results I

Theorem (Guirao, Zizler, M.) X with countable James boundary. Then ∃ | · | C∞-smooth, all x ∈ SX unpreserved extreme.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

The main results I

Theorem (Guirao, Zizler, M.) X with countable James boundary. Then ∃ | · | C∞-smooth, all x ∈ SX unpreserved extreme. Observe | · | strictly convex.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

The main results I

Theorem (Guirao, Zizler, M.) X with countable James boundary. Then ∃ | · | C∞-smooth, all x ∈ SX unpreserved extreme. Observe | · | strictly convex.

X

BX∗∗

all x ∈ SX unpreserved

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

The main results I

Theorem (Guirao, Zizler, M.) X with countable James boundary. Then ∃ | · | C∞-smooth, all x ∈ SX unpreserved extreme. Observe | · | strictly convex.

X

BX∗∗

all x ∈ SX unpreserved

Problems 1.-Characterize spaces with this property

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

The main results I

Theorem (Guirao, Zizler, M.) X with countable James boundary. Then ∃ | · | C∞-smooth, all x ∈ SX unpreserved extreme. Observe | · | strictly convex.

X

BX∗∗

all x ∈ SX unpreserved

Problems 1.-Characterize spaces with this property 2.-Can the extreme points of BX ∗∗ can be pushed far?

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

The main results II

Fabian, Whitfield, and Zizler for strongly exposed points (1983). Theorem ([Guirao, Zizler, M.) X infinite dimensional, C2-smooth.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

The main results II

Fabian, Whitfield, and Zizler for strongly exposed points (1983). Theorem ([Guirao, Zizler, M.) X infinite dimensional, C2-smooth. If ∃ x0 strongly extreme of BX,

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

The main results II

Fabian, Whitfield, and Zizler for strongly exposed points (1983). Theorem ([Guirao, Zizler, M.) X infinite dimensional, C2-smooth. If ∃ x0 strongly extreme of BX, then X superreflexive.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

The main results II

Fabian, Whitfield, and Zizler for strongly exposed points (1983). Theorem ([Guirao, Zizler, M.) X infinite dimensional, C2-smooth. If ∃ x0 strongly extreme of BX, then X superreflexive.

x0 x0 + h x0 − h BX

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

The main results II

Fabian, Whitfield, and Zizler for strongly exposed points (1983). Theorem ([Guirao, Zizler, M.) X infinite dimensional, C2-smooth. If ∃ x0 strongly extreme of BX, then X superreflexive.

x0 x0 + h x0 − h BX

f(h) := x +h+x −h−2, h < δ.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

The main results II

Fabian, Whitfield, and Zizler for strongly exposed points (1983). Theorem ([Guirao, Zizler, M.) X infinite dimensional, C2-smooth. If ∃ x0 strongly extreme of BX, then X superreflexive.

x0 x0 + h x0 − h BX

f(h) := x +h+x −h−2, h < δ. bump with uniformly continuous derivative

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Main results III

Theorem (Guirao, Zizler, M.) X separable infinite-dimensional polyhedral. Then ∃ | · | all x ∈ S(|·|) preserved extreme non-strongly extreme.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Main results III

Theorem (Guirao, Zizler, M.) X separable infinite-dimensional polyhedral. Then ∃ | · | all x ∈ S(|·|) preserved extreme non-strongly extreme. Corollary X separable infinite-dimensional polyhedral.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Main results III

Theorem (Guirao, Zizler, M.) X separable infinite-dimensional polyhedral. Then ∃ | · | all x ∈ S(|·|) preserved extreme non-strongly extreme. Corollary X separable infinite-dimensional polyhedral. Then ∃ | · | no x ∈ S|·| of continuity.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Remarks

1.-[Moltó–Orihuela–Troyanski–Valdivia’01] James space J, then J∗∗ has | · | all points in SX strongly extreme.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Remarks

1.-[Moltó–Orihuela–Troyanski–Valdivia’01] James space J, then J∗∗ has | · | all points in SX strongly extreme. 2.-[Hu–Moors–Smith’97] ℓ∞ no norm all SX preserved extreme.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Remarks

1.-[Moltó–Orihuela–Troyanski–Valdivia’01] James space J, then J∗∗ has | · | all points in SX strongly extreme. 2.-[Hu–Moors–Smith’97] ℓ∞ no norm all SX preserved extreme. 3.-[Hájek’98] Γ uncountable, then no C2 rotund norm on c0(Γ).

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Some non-separable results

Theorem (Guirao, M., Zizler) X WCG Banach, c0 ֒ → X.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Some non-separable results

Theorem (Guirao, M., Zizler) X WCG Banach, c0 ֒ → X. Then ∃| · | one-direction-uniformly (M),

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Some non-separable results

Theorem (Guirao, M., Zizler) X WCG Banach, c0 ֒ → X. Then ∃| · | one-direction-uniformly (M), i.e., all x ∈ S(X,|·|) extreme, all unpreserved, one direction, same δ.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Some non-separable results

Theorem (Guirao, M., Zizler) X WCG Banach, c0 ֒ → X. Then ∃| · | one-direction-uniformly (M), i.e., all x ∈ S(X,|·|) extreme, all unpreserved, one direction, same δ. Remark Valid for WLD spaces.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Some non-separable results

Theorem (Guirao, M., Zizler) X WCG Banach, c0 ֒ → X. Then ∃| · | one-direction-uniformly (M), i.e., all x ∈ S(X,|·|) extreme, all unpreserved, one direction, same δ. Remark Valid for WLD spaces. Proof X = G ⊕ Y, where c0 ֒ → Y, Y separable.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Some non-separable results

Theorem (Guirao, M., Zizler) X WCG Banach, c0 ֒ → X. Then ∃| · | one-direction-uniformly (M), i.e., all x ∈ S(X,|·|) extreme, all unpreserved, one direction, same δ. Remark Valid for WLD spaces. Proof X = G ⊕ Y, where c0 ֒ → Y, Y separable. By Sobczyk, take Y = c0.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Some non-separable results

Theorem (Guirao, M., Zizler) X WCG Banach, c0 ֒ → X. Then ∃| · | one-direction-uniformly (M), i.e., all x ∈ S(X,|·|) extreme, all unpreserved, one direction, same δ. Remark Valid for WLD spaces. Proof X = G ⊕ Y, where c0 ֒ → Y, Y separable. By Sobczyk, take Y = c0. ∃y∗∗

0 ∈ Y ∗∗, | · | in c0, |y ± δy∗∗ 0 | ≤ 1, all y ∈ S(Y,|·|),

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Some non-separable results

Theorem (Guirao, M., Zizler) X WCG Banach, c0 ֒ → X. Then ∃| · | one-direction-uniformly (M), i.e., all x ∈ S(X,|·|) extreme, all unpreserved, one direction, same δ. Remark Valid for WLD spaces. Proof X = G ⊕ Y, where c0 ֒ → Y, Y separable. By Sobczyk, take Y = c0. ∃y∗∗

0 ∈ Y ∗∗, | · | in c0, |y ± δy∗∗ 0 | ≤ 1, all y ∈ S(Y,|·|),

∃ T : Y → ℓ2 1 − 1, T ∗∗y∗∗

0 = 0.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Some non-separable results

Theorem (Guirao, M., Zizler) X WCG Banach, c0 ֒ → X. Then ∃| · | one-direction-uniformly (M), i.e., all x ∈ S(X,|·|) extreme, all unpreserved, one direction, same δ. Remark Valid for WLD spaces. Proof X = G ⊕ Y, where c0 ֒ → Y, Y separable. By Sobczyk, take Y = c0. ∃y∗∗

0 ∈ Y ∗∗, | · | in c0, |y ± δy∗∗ 0 | ≤ 1, all y ∈ S(Y,|·|),

∃ T : Y → ℓ2 1 − 1, T ∗∗y∗∗

0 = 0.

Put |(g, y)| = max{g, |y|}.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Some non-separable results

Theorem (Guirao, M., Zizler) X WCG Banach, c0 ֒ → X. Then ∃| · | one-direction-uniformly (M), i.e., all x ∈ S(X,|·|) extreme, all unpreserved, one direction, same δ. Remark Valid for WLD spaces. Proof X = G ⊕ Y, where c0 ֒ → Y, Y separable. By Sobczyk, take Y = c0. ∃y∗∗

0 ∈ Y ∗∗, | · | in c0, |y ± δy∗∗ 0 | ≤ 1, all y ∈ S(Y,|·|),

∃ T : Y → ℓ2 1 − 1, T ∗∗y∗∗

0 = 0.

Put |(g, y)| = max{g, |y|}. ∃S : G → c0(Γ) 1 − 1.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Some non-separable results

Theorem (Guirao, M., Zizler) X WCG Banach, c0 ֒ → X. Then ∃| · | one-direction-uniformly (M), i.e., all x ∈ S(X,|·|) extreme, all unpreserved, one direction, same δ. Remark Valid for WLD spaces. Proof X = G ⊕ Y, where c0 ֒ → Y, Y separable. By Sobczyk, take Y = c0. ∃y∗∗

0 ∈ Y ∗∗, | · | in c0, |y ± δy∗∗ 0 | ≤ 1, all y ∈ S(Y,|·|),

∃ T : Y → ℓ2 1 − 1, T ∗∗y∗∗

0 = 0.

Put |(g, y)| = max{g, |y|}. ∃S : G → c0(Γ) 1 − 1. Put η(g, y) := |(g, y)| + (S(g)2

D + T(y)2 2)1/2.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Some nonseparable results

∃ nonseparable X, c0 ֒ → X, no equivalent | · | (M). Example ℓ∞/c0. It has not strictly convex norm (Bourgain)

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

Some nonseparable results

An extension Theorem G strictly convex. Then G ⊕ c0 has | · |, all x ∈ S(X,|·|) extreme, all unpreserved, one direction, uniform.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A special property of c0(Γ)

Theorem ∃ | · | in c0(Γ), Fréchet differentiable, all x ∈ S(X,|·|) extreme, all unpreserved.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A special property of c0(Γ)

Theorem ∃ | · | in c0(Γ), Fréchet differentiable, all x ∈ S(X,|·|) extreme, all unpreserved.

• Proof. On c0(Γ)

∃ · S C∞-smooth, depends locally finitely many coordinates (using the Implicit Mapping Theorem)

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A special property of c0(Γ)

Theorem ∃ | · | in c0(Γ), Fréchet differentiable, all x ∈ S(X,|·|) extreme, all unpreserved.

• Proof. On c0(Γ)

∃ · S C∞-smooth, depends locally finitely many coordinates (using the Implicit Mapping Theorem) ∃ · LUR,F LUR, Fréchet smooth.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A special property of c0(Γ)

Theorem ∃ | · | in c0(Γ), Fréchet differentiable, all x ∈ S(X,|·|) extreme, all unpreserved.

• Proof. On c0(Γ)

∃ · S C∞-smooth, depends locally finitely many coordinates (using the Implicit Mapping Theorem) ∃ · LUR,F LUR, Fréchet smooth. T : c0(Γ) → c0(Γ \ N) ⊕ ℓ2 (a special T that annihilates some elements in ℓ∞)

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results

A special property of c0(Γ)

Theorem ∃ | · | in c0(Γ), Fréchet differentiable, all x ∈ S(X,|·|) extreme, all unpreserved.

• Proof. On c0(Γ)

∃ · S C∞-smooth, depends locally finitely many coordinates (using the Implicit Mapping Theorem) ∃ · LUR,F LUR, Fréchet smooth. T : c0(Γ) → c0(Γ \ N) ⊕ ℓ2 (a special T that annihilates some elements in ℓ∞) |(y, x)| = (y, x)S + (y2

LUR,F + x2 2)1/2, y ∈ c0(Γ \ N),

x ∈ c0.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Appendix References

References I

• M. Fabian, P

. Habala, P . Hájek, V. Montesinos, V. Zizler. Banach Space Theory: the Basis for Linear and Non-Linear Analysis Springer-Verlag, New York, 2011.

• A. J. Guirao, V. Montesinos, V. Zizler.

Preserved and unpreserved extreme points. Springer-Verlag, to appear. P . Morris Dissapearance of extreme points

• Proc. Amer. Math. Soc. 88, 2 (1983), 244–246.
• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points

Appendix References

References II

P . Hájek, V. Montesinos, and V. Zizler Geometry and Gâteaux smoothness in separable Banach spaces

• Oper. Matrices 6 (2012), (2), 201–232.
• A. J. Guirao, V. Montesinos, and V. Zizler

A note on extreme points of C∞-smooth balls in polyhedral spaces To appear.

• A. J. Guirao, V. Montesinos, and V. Zizler

Preserved and unpreserved extreme points