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Another approach to characterizations of generalized triangle inequalities in normed spaces

100%

Izumida T., Mitani K.-I., Saito K.-S.

Open Mathematics

2014

tom 12

nr 11

1615-1623

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].

Characterization of intermediate values of the triangle inequality II

88%

Sano H., Izumida T., Mitani K.-I., Ohwada T., Saito K.-S.

Open Mathematics

2014

tom 12

nr 5

778-786

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.

Ograniczanie wyników

2 Open Mathematics

2 Izumida T.

2 Mitani K.-I.

2 Saito K.-S.

1 Ohwada T.

1 Sano H.

2 2014

Wyniki wyszukiwania

Another approach to characterizations of generalized triangle inequalities in normed spaces

Characterization of intermediate values of the triangle inequality II