On the generalized triangle inequality for quasimetrics induced by noncommuting vector fields

2012 ◽  
Vol 22 (2) ◽  
pp. 95-114 ◽  
Author(s):  
A. V. Greshnov
Keyword(s):  
Vector Fields ◽  
Triangle Inequality ◽  
Generalized Triangle Inequality

scholarly journals More on reverse triangle inequality in inner product spaces

2005 ◽  
Vol 2005 (18) ◽  
pp. 2883-2893 ◽  
Author(s):  
A. H. Ansari ◽  
M. S. Moslehian
Keyword(s):  
Product Space ◽  
Triangle Inequality ◽  
Schwarz Inequality ◽  
Unit Vector ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Generalized Triangle Inequality

Refining some results of Dragomir, several new reverses of the generalized triangle inequality in inner product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that ifais a unit vector in a real or complex inner product space(H;〈.,.〉),r,s>0,p∈(0,s],D={x∈H,‖rx−sa‖≤p},x1,x2∈D−{0}, andαr,s=min{(r2‖xk‖2−p2+s2)/2rs‖xk‖:1≤k≤2}, then(‖x1‖‖x2‖−Re〈x1,x2〉)/(‖x1‖+‖x2‖)2≤αr,s.

scholarly journals Characterization of a generalized triangle inequality in normed spaces

2012 ◽  
Vol 75 (2) ◽  
pp. 735-741 ◽  
Author(s):  
Farzad Dadipour ◽  
Mohammad Sal Moslehian ◽  
John M. Rassias ◽  
Sin-Ei Takahasi
Keyword(s):  
Triangle Inequality ◽  
Normed Spaces ◽  
Generalized Triangle Inequality

Another approach to characterizations of generalized triangle inequalities in normed spaces

2014 ◽  
Vol 12 (11) ◽  
Author(s):  
Tamotsu Izumida ◽  
Ken-Ichi Mitani ◽  
Kichi-Suke Saito
Keyword(s):  
Banach Spaces ◽  
Normed Space ◽  
Triangle Inequality ◽  
Normed Spaces ◽  
Direct Sums ◽  
Nonlinear Anal ◽  
Generalized Triangle Inequality

AbstractIn 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].

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