A natural correspondence between quasiconcave functions and fuzzy norms

Abstract

In this note we show that the usual notion of fuzzy norm defined on a linear space is equivalent to that of quasiconcave function, in the sense that every fuzzy norm N:X×R→[0,1] defined on a (real or complex) linear space X is uniquely determined by a quasiconcave function f:X→[0,1]. We explore the minimum requirements that we need to impose to some quasiconcave function f:X→[0,1] in order to define a fuzzy norm N:X×R→[0,1]. Later we use this equivalence to prove some properties of fuzzy norms, like a generalisation of the celebrated Decomposition Theorem.