Characterization of a generalized triangle inequality in normed spaces

For a normed linear space $(X,\mathrm{\mid }\cdot \mathrm{\mid })$ and $p>0$ we characterize all $n$-tuples $({\mu }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\mu }_n)\in {\mathbb{R}}^n$ for which the generalized triangle inequality of the second type $$\mathrm{\parallel }{x}_1+\mathrm{.}\mathrm{.}\mathrm{.}+x_n{\mathrm{\parallel }}^p\le \frac{\mathrm{\mid }{x}_1{\mathrm{\mid }}^p}{{\mu }_1}+\mathrm{.}\mathrm{.}\mathrm{.}+\frac{\mathrm{\mid }{x}_n{\mathrm{\mid }}^p}{{\mu }_n}$$ holds for any $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n\in X$. We also characterize $({\mu }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\mu }_n)\in {\mathbb{R}}^n$ for which the reverse of the inequality above holds.