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In this paper we present some new inequalities in normed linear spaces which much improve the triangle inequality. Our results refine and generalize the corresponding ones obtained by Mitani et al. [ On sharp triangle inequalities in Banach spaces, J. Math. Anal. Appl. 336 (2007) 1178-1186]. Mathematics subject classification (2010): Primary 47A30; Secondary 26D20.

Some new bounds for the generalized triangle inequality