Multi-dimensional volumes and moduli of convexity in banach spaces

Geremia, Raymond; Sullivan, Francis

doi:10.1007/bf01811725

Cited by 21 publications

(6 citation statements)

References 14 publications

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“…Using the properties of the operators <2**_ 2 > k> l,it is easy to show that Fj(f n ) -> \\Fj\\ = 1 for eachj, 1 <./ < j fc + 1, and H^ + .F 2 + • • • + F fc+1 || = k + 1. This leads to the following generalization of a theorem by Dixmier (1948;Theoreme 20) which appears in Geremia and Sullivan (1981 is a linearly dependent set for some k > 1, then F = x for some x e S(E).…”

Section: If E* Is K-smooth Then E Is K-rotund 2 If E* Is K-rotundmentioning

confidence: 91%

“…[See Silverman (1951) and Sullivan (1979).] The following lemma appears in Geremia and Sullivan (1981;page 233).…”

Section: A:-weak Rotundity and ^-Smoothnessmentioning

confidence: 99%

“…has generalized uniform rotundity by using the idea of "area" given in Silverman (1951) and Geremia and Sullivan (1981). In this paper I will also use this idea of "area" when generalizing weak rotundity for real and complex Banach spaces.…”

Section: Introductionmentioning

confidence: 99%

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Rotundity redux

Yorke¹

1985

J Aust Math Soc A

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Recently the concept of uniform rotundity was generalized for real Banach spaces by using a type of "area" devised for these spaces. This paper modifies the methods used for uniform rotundity and applies them to weak rotundity in real and complex spaces. This leads to the definition of A:-smoothness, A:-very smoothness and fc-strong smoothness. As an application, several sufficient conditions for reflexivity are obtained.

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Section: If E* Is K-smooth Then E Is K-rotund 2 If E* Is K-rotundmentioning

confidence: 91%

“…[See Silverman (1951) and Sullivan (1979).] The following lemma appears in Geremia and Sullivan (1981;page 233).…”

Section: A:-weak Rotundity and ^-Smoothnessmentioning

confidence: 99%

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Rotundity redux

Yorke¹

1985

J Aust Math Soc A

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“…Another approach to finite dimensional uniform convexity was found by Sullivan [103] who defined the modulus of k-convexity. Theorem 25 was proved by Lin [74], but its partial cases with k = 1, 2 had been earlier obtained in [37] and [40]. The reader should be warned that there are different definitions of k-uniform convexity in the literature (see, for instance, [52], p. 73).…”

Section: Obviously τ Cs(x) = Inf Limmentioning

confidence: 99%

Geometrical Background of Metric Fixed Point Theory

Prus

2001

Handbook of Metric Fixed Point Theory

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“…For our extensions we make use of the notion of "k-dimensional volume" as discussed in [3] and [14].…”

Section: Preliminariesmentioning

confidence: 99%

A note on Picard iterates of nonexpansive mappings

Kim

Kirk

2001

Ann. Polon. Math.

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Let X be a Banach space, C a closed subset of X, and T : C → C a nonexpansive mapping. It has recently been shown that if X is reflexive and locally uniformly convex and if the fixed point set F (T) of T has nonempty interior then the Picard iterates of the mapping T always converge to a point of F (T). In this paper it is shown that if T is assumed to be asymptotically regular, this condition can be weakened much further. Finally, some observations are made about the geometric conditions imposed.

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Multi-dimensional volumes and moduli of convexity in banach spaces

Cited by 21 publications

References 14 publications

Rotundity redux

Rotundity redux

Geometrical Background of Metric Fixed Point Theory

A note on Picard iterates of nonexpansive mappings

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