A natural correspondence between quasiconcave functions and fuzzy norms
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909090856304640 |
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| author | Sánchez, Javier Cabello González, Daniel Morales |
| author_facet | Sánchez, Javier Cabello González, Daniel Morales |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01283 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A natural correspondence between quasiconcave functions and fuzzy norms Sánchez, Javier Cabello González, Daniel Morales Metric Geometry 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. |
| title | A natural correspondence between quasiconcave functions and fuzzy norms |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2402.01283 |