Characterization of a generalized triangle inequality in normed spaces

Abstract

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.