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A new geometrical constant of Banach spaces and the uniform normal structure

100%
Mitani K.-I., Saito K.-S.
Commentationes Mathematicae
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2009
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tom 49
|
nr 1
EN
We introduce and study a new geometrical constant \(\gamma_{X,\psi\) of a Banach space \(X\), by using the notion of \(\psi\)-direct sum given in [Y. Takahashi, M. Kato and K.-S. Saito, J. Inequal. Appl. 7 (2002), 179-186]. At first, we characterize uniform non-squareness in terms of \(\gamma_{X,\psi}\). Moreover, we consider Banach spaces having uniform normal structure.
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Another approach to characterizations of generalized triangle inequalities in normed spaces

81%
Izumida T., Mitani K.-I., Saito K.-S.
Open Mathematics
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2014
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tom 12
|
nr 11
1615-1623
EN
In this paper, we consider a generalized triangle inequality of the following type: $$\left\| {x_1 + \cdots + x_n } \right\|^p \leqslant \frac{{\left\| {x_1 } \right\|^p }} {{\mu _1 }} + \cdots + \frac{{\left\| {x_2 } \right\|^p }} {{\mu _n }}\left( {for all x_1 , \ldots ,x_n \in X} \right),$$ where (X, ‖·‖) is a normed space, (µ1, ..., µn) ∈ ℝn and p > 0. By using ψ-direct sums of Banach spaces, we present another approach to characterizations of the above inequality which is given by [Dadipour F., Moslehian M.S., Rassias J.M., Takahasi S.-E., Nonlinear Anal., 2012, 75(2), 735–741].
3
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On the calculation of the Dunkl-Williams constant of normed linear spaces

81%
Mizuguchi H., Saito K.-S., Tanaka R.
Open Mathematics
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2013
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tom 11
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nr 7
1212-1227
EN
Recently, Jiménez-Melado et al. [Jiménez-Melado A., Llorens-Fuster E., Mazcuñán-Navarro E.M., The Dunkl-Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl., 2008, 342(1), 298–310] defined the Dunkl-Williams constant DW(X) of a normed linear space X. In this paper we present some characterizations of this constant. As an application, we calculate DW(ℓ2-ℓ∞) in the Day-James space ℓ2-ℓ∞.
4
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Characterization of intermediate values of the triangle inequality II

71%
Sano H., Izumida T., Mitani K.-I., Ohwada T., Saito K.-S.
Open Mathematics
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2014
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tom 12
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nr 5
778-786
EN
In [Mineno K., Nakamura Y., Ohwada T., Characterization of the intermediate values of the triangle inequality, Math. Inequal. Appl., 2012, 15(4), 1019–1035] there was established a norm inequality which characterizes all intermediate values of the triangle inequality, i.e. C n that satisfy 0 ≤ C n ≤ Σj=1n ‖x j‖ − ‖Σj=1n x j‖, x 1,...,x n ∈ X. Here we study when this norm inequality attains equality in strictly convex Banach spaces.
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