Characterization of p-Banach Spaces Based on a Reverse Triangle Inequality

Abstract

In this paper, we deal with the reverse of the generalized triangle inequality of the second type in quasi-Banach spaces. More exactly, by using the concept of equivalent p-norms, we provide some necessary and sufficient conditions for n-tuples to satisfy the mentioned inequality. As applications, we improve some already known results and present some characterizations of p-Banach spaces among quasi-Banach spaces. In particular, we prove that a quasi-Banach space such as X is a p-Banach space if and only if for all (μ1, ⋯ , μn) ∈ Rn satisfying μj> 0 for some j and μi< 0 for all i≠ j , the generalized triangle inequality of the second type ∑i=1n‖xi‖pμi≤‖∑i=1nxi‖p(xi∈X) holds only with the assumption μj≥ max i∈{1,⋯,n}\{j}{ 1 , | μi| } .