We investigate a generalized triangle inequality of the second type in the framework of quasi normed spaces. More precisely, by using the well-known Aoki-Rolewicz theorem and some quasi normed inequalities, we obtain some regions of R n which contain the set of all ntuples satisfying the mentioned inequality. Moreover, some reverse inclusions are also discussed. As applications, we deduce some new results associated with generalizations of the triangle inequality in p -normed spaces and we get some already known results in a new approach. Mathematics subject classification (2010): 46A16, 47A30, 46B20.
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