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 $$\Vert x_1+...+x_n{\Vert }^p\le \frac{|x_1{|}^p}{{\mu }_1}+...+\frac{|x_n{|}^p}{{\mu }_n}$$ 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.

11 Pages, to appear in Nonlinear Anal-TMA

References in corpus (2)

• Reverses of the Triangle Inequality in Banach Spaces
• Reverse triangle inequality in Hilbert $C^{\ast }$-modules