Ken-Ichi Mitani

On Sharp Triangle Inequalities in Banach Spaces II

Sharp triangle inequality and its reverse in Banach spaces were recently showed by Mitani et al. (2007). In this paper, we present equality attainedness for these inequalities in strictly convex Banach spaces.

Another Aspect of Triangle Inequality

We introduce the notion of 𝜓-norm by considering the fact that an absolute normalized norm on ℂ2 corresponds to a continuous convex function 𝜓 on the unit interval [0,1] with some conditions. This is a generalization of the notion of 𝑞-norm introduced by Belbachir et al. (2006). Then we show that a 𝜓-norm is a norm in the usual sense.

Extremal structure of absolute normalized norms on R2 and the James constant

Abstract We denote by AN2 the set of all absolute normalized norms on R 2 . The set AN2 has a convex structure with respect to the usual operation. In this paper we calculate the James constant of ( R 2 , ‖ · ‖ ) when ∥·∥ is an extreme point of AN2.

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. Cn that satisfy 0 ≤ Cn ≤ Σj=1n ‖xj‖ − ‖Σj=1nxj‖, x1,...,xn ∈ X. Here we study when this norm inequality attains equality in strictly convex Banach spaces.

Another approach to characterizations of generalized triangle inequalities in normed spaces

AbstractIn this paper, we consider a generalized triangle inequality of the following type:

Dual of two dimensional Lorentz sequence spaces

\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),

Smoothness of ψ-direct sums of Banach spaces

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].