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arXiv · 2402.01283

A natural correspondence between quasiconcave functions and fuzzy norms

Javier Cabello Sánchez·Daniel Morales González
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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 $N:X\times\mathbb{R}[0,1]$ 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.

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Javier Cabello Sánchez, Daniel Morales González. 2024-02-02. A natural correspondence between quasiconcave functions and fuzzy norms. https://doi.org/10.1016/j.fss.2022.10.005

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