A natural correspondence between quasiconcave functions and fuzzy norms

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 $N:X\times\mathbb{R}[0,1]$ defined on a (real or complex) linear space X is uniquely determined by a quasiconcave function $f:X\to[0, 1]$. We explore the minimum requirements that we need to impose to some quasiconcave function $f:X\to[0, 1]$ in order to define a fuzzy norm $N:X\times\mathbb{R}[0,1]$. Later we use this equivalence to prove some properties of fuzzy norms, like a generalisation of the celebrated Decomposition Theorem.