On uniformly convex functions

Guillaume, Grelier,; Raja, M.

doi:10.1016/j.jmaa.2021.125442

Cited by 4 publications

(7 citation statements)

References 29 publications

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“…In order to state the results from our paper [30] that we will need later, it is necessary to introduce a certain number of quantities related to sets in Banach spaces. Let us denote by H the set of all the open half-spaces of X, that is, all the sets of the form H = {x ∈ X, x * (x) > α}, with x * ∈ X * and α ∈ R. A slice of of D ⊂ X is a set of the form D ∩ H = ∅, where H ∈ H.…”

Section: Different Ways To Quantify Swcmentioning

confidence: 99%

“…( 3), ( 4) and ( 5) follow from set identities: (A ∪ B) U = A U ∪ B U , (λA) U = λA U and (A + B) U = A U + B U . Statement (6), the most tricky, was proved in [30,Theorem 6.7].…”

Section: Theorem 34 ([30]mentioning

confidence: 99%

“…Take Γ(T ) = ε ′ < ε and 1 < λ < ε/ε ′ . By Theorem 3.4, the set B = λT (B X ) supports a convex bounded ε-uniformly convex function f that we may assume it is also Lipschitz, see [30,Proposition 5.4]. The function f • T is ε-uniformly convex with respect to the pseudometric d(x, y) = T (x) − T (y) on λB X .…”

Section: Quantifying Uniform Convexity For Operatorsmentioning

confidence: 99%

“…The class of SWC sets lies strictly between the norm compact and the weakly compact subsets. The theory of SWC sets has been developed during the last 15 years in a series of papers [43,14,15,44,50,16,47,48,38,30].…”

Section: Introductionmentioning

confidence: 99%

“…[30]). Let (X, • ) be a Banach space, let f : X → [0, +∞] be a proper convex function and let C ⊂ dom(f ) be a bounded convex set.…”

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confidence: 99%

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Applications of the quantification of super weak compactness

Guillaume¹,

Raja²

2022

Preprint

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We introduce a measure of super weak noncompactness Γ defined for bounded linear operators and subsets in Banach spaces that allows to state and prove a characterization of the Banach spaces which are subspaces of a Hilbert generated space. The use of super weak compactness and Γ casts light on the structure of these Banach spaces and complements the work of Argyros, Fabian, Farmaki, Godefroy, Hájek, Montesinos, Troyanski and Zizler on this subject. A particular kind of relatively super weakly compact sets, namely uniformly weakly null sets, plays an important role and exhibits connections with Banach-Saks type properties.

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Section: Different Ways To Quantify Swcmentioning

confidence: 99%

“…( 3), ( 4) and ( 5) follow from set identities: (A ∪ B) U = A U ∪ B U , (λA) U = λA U and (A + B) U = A U + B U . Statement (6), the most tricky, was proved in [30,Theorem 6.7].…”

Section: Theorem 34 ([30]mentioning

confidence: 99%

Section: Quantifying Uniform Convexity For Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…[30]). Let (X, • ) be a Banach space, let f : X → [0, +∞] be a proper convex function and let C ⊂ dom(f ) be a bounded convex set.…”

mentioning

confidence: 99%

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Applications of the quantification of super weak compactness

Guillaume¹,

Raja²

2022

Preprint

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Extremal structure in ultrapowers of Banach spaces

García-Lirola

Guillaume

Zoca

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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Given a bounded convex subset C of a Banach space X and a free ultrafilter $${\mathcal {U}}$$ U , we study which points $$(x_i)_{\mathcal {U}}$$ ( x i ) U are extreme points of the ultrapower $$C_{\mathcal {U}}$$ C U in $$X_{\mathcal {U}}$$ X U . In general, we obtain that when $$\{x_i\}$$ { x i } is made of extreme points (respectively denting points, strongly exposed points) and they satisfy some kind of uniformity, then $$(x_i)_{\mathcal {U}}$$ ( x i ) U is an extreme point (respectively denting point, strongly exposed point) of $$C_\mathcal U$$ C U . We also show that every extreme point of $$C_{{\mathcal {U}}}$$ C U is strongly extreme, and that every point exposed by a functional in $$(X^*)_{{\mathcal {U}}}$$ ( X ∗ ) U is strongly exposed, provided that $$\mathcal U$$ U is a countably incomplete ultrafilter. Finally, we analyse the extremal structure of $$C_{\mathcal {U}}$$ C U in the case that C is a super weakly compact or uniformly convex set.

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Ergodicity and super weak compactness

Grelier,

Raja

2024

Ann. Funct. Anal.

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On uniformly convex functions

Cited by 4 publications

References 29 publications

Applications of the quantification of super weak compactness

Applications of the quantification of super weak compactness

Extremal structure in ultrapowers of Banach spaces

Ergodicity and super weak compactness

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