Characterization of a generalized triangle inequality in normed spaces

Details Characterization of a generalized triangle inequality in normed spaces Characterization of a generalized triangle inequality in normed spaces

2011-09-08

arxiv.org/abs/1109.1773v1

Nonlinear Anal-TMA 75 (2012), no. 2, 735-741

Mathematics

Functional Analysis

11 Pages, to appear in Nonlinear Anal-TMA

Scientific paper

For a normed linear space $(X,|\cdot|)$ and $p>0$ we characterize all $n$-tuples $(\mu_1,...,\mu_n)\in\mathbb{R}^{n}$ for which the generalized triangle inequality of the second type $$\|x_1+...+x_n\|^p\leq\frac{|x_1|^p}{\mu_1}+...+\frac{|x_n|^p}{\mu_n}$$ holds for any $x_1,...,x_n\in X$. We also characterize $(\mu_1,...,\mu_n)\in\mathbb{R}^{n}$ for which the reverse of the inequality above holds.

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