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Another approach to characterizations of generalized triangle inequalities in normed spaces

Tamotsu IzumidaKen-Ichi MitaniKichi-Suke Saito — 2014

Open Mathematics

In this paper, we consider a generalized triangle inequality of the following type: x 1 + + x n p x 1 p μ 1 + + x 2 p μ n f o r a l l x 1 , ... , x n X , 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

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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