On geometry of cones

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On geometry of cones

Fernando García Castaño^a

(Joint work with M. A. Melguizo Padial^aand V. Montesinos^b)

aDepartment of Applied Mathematics University of Alicante

bInstituto de Matemática Pura y Aplicada Universitat Politècnica de València

Workshop on Functional Analysis Valencia 2015 on the occasion of the 60th birthday of José Bonet

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Contents

1 Introduction

2 Main results

3 Consequences

4 Bibliography

Fernando García (U. Alicante) On geometry of cones WFA, Valencia 2015 2 / 30

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Introduction

Notation and terminology X denotesa normed space

An open half space ofX is a set{x ∈X:f(x)< λ}for some f ∈X^∗\ {0_X^∗}andλ∈R. We denote it briefly by{f < λ}

An slice of a setCis a non empty intersection ofCwith an open half space ofX

f =λ S={f < λ} ∩C C

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Introduction

Notation and terminology X denotes a normed space

Anopen half spaceofX is a set{x ∈X:f(x)< λ}for some f ∈X^∗\ {0_X^∗}andλ∈R. We denote it briefly by{f < λ}

An slice of a setCis a non empty intersection ofCwith an open half space ofX

f =λ S={f < λ} ∩C C

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Introduction

Notation and terminology X denotes a normed space

An open half space ofX is a set{x ∈X:f(x)< λ}for some f ∈X^∗\ {0_X^∗}andλ∈R. We denote it briefly by{f < λ}

Ansliceof a setCis a non empty intersection ofCwith an open half space ofX

f=λ S={f < λ} ∩C C

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Introduction

Denting points

Definition

LetCbe a subset ofX,c ∈C is said to be adenting point ofCif c 6∈conv(C\B_ε(c)),∀ε >0.

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Introduction

Denting points

Definition

LetCbe a subset ofX,c ∈C is said to be a denting point ofCif c 6∈conv(C\B_ε(c)),∀ε >0.

C

c

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Introduction

Denting points

Definition

C

c Bǫ(c)

ǫ

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Introduction

Denting points

Definition

C\Bǫ(c)

c

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Introduction

Denting points

Definition

c

conv(C\B_ǫ(c))

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Introduction

Denting points

c is a denting point ofC if and only if∀ε >0⇒ ∃S_ε⊂C, (slice)c ∈S_ε withdiamS_ε≤ε

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Introduction

Denting points

c is a denting point ofC if and only if∀ε >0⇒ ∃S_ε⊂C, (slice)c ∈S_ε with diamS_ε≤ε

C

c

ǫ

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Introduction

Denting points

C

c

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Introduction

Denting points

C

c

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Introduction

Denting points

C

c

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Introduction

Points of continuity

Definition

LetCbe a subset ofX,c ∈C is said to be apoint of continuityforCif the identity map(C,weak)→(C,k k)is continuous atc.

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Introduction

c is a point of continuity forCif and only iffor everyopen ballB_ε(c), there existsa weakly openU such that

c∈U∩C⊂B_ε(c)∩C

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Introduction

c is a point of continuity forCif and only if for every open ballB_ε(c), there exists a weakly openU such that

C

c

ǫ

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Introduction

C

c

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Introduction

C

c

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Introduction

Denting points and points of continuity

denting point⇒point of continuity

Definition

c is an extremal point ofCif it does not belong to any non degenerate line segment inC

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Introduction

denting point⇒point of continuity Definition

c is anextremal point ofCif it does not belong to any non degenerate line segment inC

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Introduction

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Introduction

Extremal points

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Introduction

Theorem (Lin–Lin–Troyanski, 1985)

Letcbe anextremal pointof aclosed, convex, and bounded subsetC of aBanach space. Ifc is apoint of continuityforC, then itis a denting point.

What about cones?

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Introduction

Theorem (Lin–Lin–Troyanski, 1985)

Letcbe an extremal point of a closed, convex, and bounded subsetC of a Banach space. Ifc is a point of continuity forC, then it is a denting point.

What about cones?

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Introduction

Denting points, points of continuity, and cones Definition

1 A non empty convex subsetC ofX is called aconeif

αC⊂C,∀α≥0

2 A coneCis called pointed ifC∩(−C) ={0_X}

0X

C

c αc α^′c

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Introduction

1 A non empty convex subsetC ofX is called a cone if

αC⊂C,∀α≥0

2 A coneC is calledpointedifC∩(−C) ={0_X}

0_X

C

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Introduction

1 A non empty convex subsetC ofX is called a cone if

αC⊂C,∀α≥0

2 A coneC is called pointed ifC∩(−C) ={0_X}

0_X -C C

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Introduction

Denting points, points of continuity, and cones

Problem (Gong, 1995)

The property ofpoint of continuityat theoriginfor aclosedand pointed conein a normed space,is really weaker thanthe property ofdenting pointat the origin of the cone?

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Introduction

The property of point of continuity at the origin for a closed and pointed cone in a normed space, is really weaker than the property of denting point at the origin of the cone?

A negative answer for Banach spaces Theorem (Daniilidis, 2000)

LetCbe aclosedand pointedconein aBanach spaceX. Then 0_X is adenting pointofC if and onlyif it is apoint of continuityforC.

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Introduction

The former characterization allowed Daniilidis to prove theequivalence (into the frame of Banach spaces)between two density results of Arrow, Barankin and Blackwell’s type. One due to Petschke (1990) and

another due to Gong (1995).

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Introduction

A partial answer for non closed cones Theorem (Kountzakis–Polyrakis, 2006)

LetX be a normed space such that the set of quasi-interior positive elements ofX^∗ (qiC^∗ for short) isnon empty. Then 0_X is adenting pointof a pointed coneC if and onlyif it is apoint of continuityforC.

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Introduction

An elementf ∈C^∗ :={g∈X^∗:g(c)≥0,∀c ∈C}belongs toqiC^∗if

∪^n≥1[−nf,nf] =X^∗, where

[−nf,nf] :={g ∈X^∗: −nf(c)≤g(c)≤nf(c),∀c∈C}

Problem (Kountzakis–Polyrakis, 2006)

If the origin is a point of continuity for a cone in a normed space, is qiC^∗6=∅?

If the answer is positive, then

0_X denting point⇔point of continuity

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Introduction

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Introduction

∪^n≥1[−nf,nf]=X^∗, where

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Introduction

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Introduction

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Main results

The following example provides a positive answer to Gong’s question for non closed cones and a negative answer to Kountzakis–Polyrakis’

question.

Example

Let us defineX :=R²andC :=R×(0,+∞)∪ {(0,0)}which is a pointed cone. Then 0_X ispoint of continuityforC, it isnot a denting point, andqiC^∗ =∅.

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Main results

GivenC⊂X ⇒Ce denotes the closure ofC in(X^∗∗,weak^∗)

Theorem 1

LetX be a normed space, 0_X is a denting point of a pointed coneCif and only if it is a point of continuity forCandCe ⊂X^∗∗is pointed.

The former example shows that the assumption "Ce is pointed" cannot be dropped

IfX reflexive⇒Ce =C^(X,^weak⁾ =C^{(X,k k)} Thus, for reflexive spaces:

Theorem 1⇔Daniilidis’ Theorem

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Main results

GivenC⊂X ⇒Ce denotes the closure ofC in(X^∗∗,weak^∗) Theorem 1

LetX be a normed space, 0_X is adenting pointof a pointed coneCif and onlyif it is apoint of continuityforCandCe ⊂X^∗∗ispointed.

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Main results

The former example shows thatthe assumption "Ce is pointed" cannot be dropped

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Main results

IfX reflexive⇒Ce =C^(X,^weak⁾

=C^{(X,k k)} Thus, for reflexive spaces:

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Main results

IfX reflexive⇒Ce =C^(X,^weak⁾ =C^{(X,k k)}

Thus, for reflexive spaces:

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Main results

IfX reflexive⇒Ce =C^(X,^weak⁾ =C^{(X,k k)} Thus,for reflexive spaces:

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Main results

The proof of Theorem 1 is based on the following reformulation of Proposition 2.2 of Guirao–Montesinos–Zizler (2014).

0_X is called a weakly strongly extreme pointCif∀(cn)n,(˜cn)n inC limn

cn+ ˜cn

2 =0_X ⇒weak-limncn=0_X. Proposition 1

LetX be a normed space andC⊂X a pointed cone. Consider the following properties.

(i) 0_X is a weakly strongly extreme point ofC;

(ii) Ce ⊂X^∗∗is pointed (i.e., 0_X is a preserved extreme point ofC); (iii) For anyR >0 andC_R :=C∩B_R(0_X), the family of open slices

containing 0_X forms a neighbourhood base for 0_X relative to (C_R,weak).

Then we have (i)⇒(ii)⇒(iii). IfCis also closed, then the three properties above are equivalent.

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Main results

0_X is called aweakly strongly extreme pointCif∀(cn)n,(˜cn)n inC limn

cn+ ˜cn

2 =0_X ⇒weak-limncn=0_X.

Proposition 1

(ii) Ce ⊂X^∗∗is pointed (i.e., 0_X is a preserved extreme point ofC); (iii) For anyR >0 andC_R :=C∩B_R(0_X), the family of open slices

containing 0_X forms a neighbourhood base for 0_X relative to (C_R,weak).

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Main results

0_X is called a weakly strongly extreme pointCif∀(cn)n,(˜cn)n inC limn

cn+ ˜cn

2 =0_X ⇒weak-limncn=0_X. Proposition 1

(ii) Ce ⊂X^∗∗is pointed (i.e., 0_X is a preserved extreme point ofC);

(iii) For anyR>0 andC_R :=C∩B_R(0_X), the family of open slices containing 0_X forms a neighbourhood base for 0_X relative to (C_R,weak).

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Main results

Proof of Theorem 1: (Sketch)

0_X ∈W ∩C⊂C_ε:=B_ε(0_X)∩C

The family of open slices containing 0_X forms a neighbourhood base for 0_X relative to(C_ε,weak)

⇒ ∃f ∈X^∗ andλ >0 such that{f < λ} ∩C is bounded

⇒0_X is denting point ofC

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Main results

0_X ∈W ∩C⊂C_ε:=B_ε(0_X)∩C

The family of open slices containing 0_X forms a neighbourhood base for 0_X relative to(C_ε,weak)

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Main results

0_X ∈W ∩C⊂C_ε:=B_ε(0_X)∩C

The family of open slices containing 0_X forms a neighbourhood base for 0_X relative to(Cε,weak)

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Main results

0_X ∈W ∩C⊂C_ε:=B_ε(0_X)∩C

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Main results

0_X ∈W ∩C⊂C_ε:=B_ε(0_X)∩C

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Main results

0_X ∈W ∩C⊂C_ε:=B_ε(0_X)∩C

0_X

{f < λ}

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Main results

0_X ∈W ∩C⊂C_ε:=B_ε(0_X)∩C

{f <^λ_n}

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Main results

0_X ∈W ∩C⊂C_ε:=B_ε(0_X)∩C

0_X

{f <_n+m^λ }

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Main results

Daniilidis’ theorem and Kountzakis–Polyrakis’ theorem can be obtainedas consequences of Theorem 1

Proposition 2

LetX be a normed space andC⊂X a pointed cone. Assume that at least one of the following two conditions holds:

1 X is a Banach space,Cis closed, and 0_X is a point of continuity forC;

2 qiC^∗6=∅.

ThenCe ⊂X^∗∗is pointed.

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Main results

Daniilidis’ theorem and Kountzakis–Polyrakis’ theorem can be obtained as consequences of Theorem 1

Proposition 2

2 qiC^∗6=∅.

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Main results

Proposition 2

2 qiC^∗6=∅.

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Main results

Proposition 2

2 qiC^∗6=∅.

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Main results

C^#:={f ∈X^∗:f(c)>0,∀c∈C\ {0_X}}

⇒qiC^∗⊂C^#⊂C^∗

Theorem 2

LetX be a normed space andC⊂X a pointed cone. The following are equivalent:

1 0_X is a denting point ofC.

2 0_X is a point of continuity and a weakly strongly extreme point of C.

3 IntC^#6=∅.

4 There existf ∈S_X^∗ and 0< δ <1 such that {f ≤λ} ∩C ⊂ λ

δB_X,∀λ >0.

5 There existsf ∈S_X^∗ such that inf

SX∩Cf >0.

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Main results

C^#:={f ∈X^∗:f(c)>0,∀c∈C\ {0_X}} ⇒qiC^∗⊂C^#⊂C^∗

Theorem 2

3 IntC^#6=∅.

δB_X,∀λ >0.

SX∩Cf >0.

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Main results

C^#:={f ∈X^∗:f(c)>0,∀c∈C\ {0_X}} ⇒qiC^∗⊂C^#⊂C^∗

Theorem 2

3 IntC^#6=∅.

δB_X,∀λ >0.

SX∩Cf >0.

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Main results

C^#:={f ∈X^∗:f(c)>0,∀c∈C\ {0_X}} ⇒qiC^∗⊂C^#⊂C^∗

Theorem 2

3 IntC^#6=∅.

δB_X,∀λ >0.

SX∩Cf >0.

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Main results

C^#:={f ∈X^∗:f(c)>0,∀c∈C\ {0_X}} ⇒qiC^∗⊂C^#⊂C^∗

Theorem 2

3 IntC^#6=∅.

δB_X,∀λ >0.

SX∩Cf >0.

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Main results

C^#:={f ∈X^∗:f(c)>0,∀c∈C\ {0_X}} ⇒qiC^∗⊂C^#⊂C^∗

Theorem 2

3 IntC^#6=∅.

δB_X,∀λ >0.

SX∩Cf >0.

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Main results

C^#:={f ∈X^∗:f(c)>0,∀c∈C\ {0_X}} ⇒qiC^∗⊂C^#⊂C^∗

Theorem 2

3 IntC^#6=∅.

δB_X,∀λ >0.

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Main results

What about otherclassical characterizationsfor 0_X to be a denting point of a closed cone in a Banach space?

They remain true for non closed cones in normed spaces!

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Main results

What about otherclassical characterizationsfor 0_X to be a denting point of a closed cone in a Banach space?

Theyremain true for non closed cones in normed spaces!

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Main results

Theorem 3

2 Chas a bounded slice (Property weak (π)).

3 Chas a bounded base.

4 IntC^∗ 6=∅(C^∗ is a solid cone).

5 There existsf ∈S_X^∗ and 0< δ <1 such that

f(c)> δkc k,∀c ∈C\ {0} (Angle property).

6 0_X is a strongly exposed point forC

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Main results

Theorem 3

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Main results

Theorem 3

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Main results

Theorem 3

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Main results

Theorem 3

f(c)> δkc k,∀c∈C\ {0} (Angle property).

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Main results

Theorem 3

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Main results

Theorem 3

3 Chas abounded base.

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Consequences

Definition

A non emptyconvex subsetBof a coneCis called abaseforC, if 0_X 6∈Band each elementc∈C\ {0}has a unique representation of the formc=λb, withλ >0 andb∈B.

A baseBis called a bounded base if it is a bounded subset ofX. B⊂Cis a base if and only∃f ∈C^#:B=f⁻¹(1)∩C

C

c=λb b

B

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Consequences

Definition

A non empty convex subsetBof a coneCis called a base forC, if 0_X 6∈Band each elementc∈C\ {0}has a unique representation of the formc=λb, withλ >0 andb∈B.

A baseB is called abounded baseif it is a bounded subset ofX.

B⊂Cis a base if and only∃f ∈C^#:B=f⁻¹(1)∩C

0_X

C

c=λb b

B

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Consequences

Definition

A non empty convex subsetBof a coneCis called a base forC, if 0_X 6∈Band each elementc∈C\ {0}has a unique representation of the formc=λb, withλ >0 andb∈B.

A baseB is called a bounded base if it is a bounded subset ofX. B⊂C is a base if and only∃f ∈C^#:B=f⁻¹(1)∩C

C

c=λb b

B

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Consequences

Corollary 1

LetX be a normed space andCa pointed cone. The following statements hold.

1 IfChas a bounded base, then every sequence inC which weakly converges to 0_X also converges to 0_X in norm. The reverse is true when qiC^∗ 6= ∅.

2 Assume that qiC^∗ 6=∅. ThenC has a base but does not have bounded base if and only ifC∩S_X has a sequence which weakly converges to 0_X.

3 Cdoes not have any base if and only if for everyf ∈X^∗there exists a non constant sequence in Kerf∩Cwhich converges to 0_X in norm.

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Consequences

Corollary 1

2 Assume that qiC^∗ 6=∅. ThenChas a base but does not have bounded base if and only ifC∩S_X has a sequence which weakly converges to 0_X.

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Consequences

Corollary 1

2 Assume that qiC^∗ 6=∅. ThenChas a base but does not have bounded base if and only ifC∩S_X has a sequence which weakly converges to 0_X.

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Consequences

Corollary 2

LetX be a normed space andCa pointed cone. If 0_X is a point of continuity forCandCe ⊂X^∗∗is pointed, then each weakly compact subset ofX has super efficient points.

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Consequences

Following the spirit of the paper of Casini and Miglierina (2010) on compact basis for cones, we obtain

Theorem 4

LetX be a Banach space andCa pointed and closed cone onX. ThenC^#⊂X^∗ is a non empty open subset if and only ifC has

bounded slices and each of them has, in fact, weakly compact closure.

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Consequences

Following the spirit of the paper of Casini and Miglierina (2010) on compact basis for cones, we obtain

Theorem 4

LetX be a Banach space andCa pointed and closed cone onX. ThenC^#⊂X^∗ is a non empty open subset if and only ifC has

bounded slices and each of them has, in fact, weakly compact closure.

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Consequences

Theorem 5

The following are equivalent for a Banach spaceX:

1 X is reflexive.

2 There exists a closed and pointed coneCwith non empty interior which has a slice with weakly compact closure.

3 IfC is any closed and pointed cone inX. Then either every slice in Cis unbounded or every slice inChas weakly compact closure.

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Consequences

Theorem 5

1 X is reflexive.

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Consequences

Theorem 5

1 X is reflexive.

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Consequences

IfC⊂X^∗ is a cone, thenC∗:={x ∈X:f(x)≥0,∀f ∈C}is calledthe polar coneforC

Corollary 3

A Banach spaceX is reflexive if and only if every closed coneC ⊂X^∗ for which 0_X^∗ is a denting point verifiesIntC∗6=∅.

Corollary 4

LetX be a reflexive Banach spaceX andC⊂X a closed cone. Then IntC 6=∅if and only if 0_X^∗ is a denting point ofC^∗.

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Consequences

IfC⊂X^∗ is a cone, thenC∗:={x ∈X:f(x)≥0,∀f ∈C}is called the polar cone forC

Corollary 3

Corollary 4

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Consequences

IfC⊂X^∗ is a cone, thenC∗:={x ∈X:f(x)≥0,∀f ∈C}is called the polar cone forC

Corollary 3

Corollary 4

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Consequences

Problem (Kountzakis–Polirakis’ question for closed cones) LetX be a normed space andC⊂X a closed and pointed cone such that 0_X is a point of continuity. Is qiC^∗ non empty?

A positive answer would imply a negative answer to Gong’s question!

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Consequences

Problem (Kountzakis–Polirakis’ question for closed cones) LetX be a normed space andC⊂X a closed and pointed cone such that 0_X is a point of continuity. Is qiC^∗ non empty?

A positive answer would imply a negative answer to Gong’s question!

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X. H. Gong, Density of the Set of Positive Proper Minimal Points in the Set of Minimal Points, Journal of Optimization Theory and Applications, 86(3) (1995) 609–630.

A. J. Guirao, V. Montesinos, V. Zizler, On Preserved and

Unpreserved Extreme Points, in: J. C. Ferrando, M. López-Pellicer (Eds.), Descriptive Topology and Functional Analysis, Springer Proceedings in Mathematics & Statistics Volume 80, Springer International Publishing, 2014, pp. 163–193.

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