Geometric interpretation of magnitude
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912682902290432 |
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| author | Asao, Yasuhiko Gomi, Kiyonori |
| author_facet | Asao, Yasuhiko Gomi, Kiyonori |
| contents | For an $n\times n$ positive definite symmetric matrix $Z$ with $Z_{ii} = 1$ for all $i$, we show that there exists a set of vectors $V_Z\subset \mathbb{R}^n$ such that the radius $R$ of the circumsphere of $V_Z$ satisfies ${\rm Mag}\ Z = (1-R^2)^{-1}$. This leads us to interpret geometrically several known and new facts on magnitude. In particular, we show that ${\rm Mag}\ Z_{X}< n$ for an $n$-point metric space $X$ of negative type with $n>1$. This result gives a negative answer to a problem given by Gomi--Meckes \cite{GM}. Furthermore, we also have a similar geometric description of magnitude for general real symmetric matrix $Z$ with $Z_{ii} = 1$ for all $i$. In this case, the radius corresponds to that of a circum-quasi-sphere, namely the set of points having a prescribed norm in a vector space endowed with an indefinite inner product. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26118 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric interpretation of magnitude Asao, Yasuhiko Gomi, Kiyonori Metric Geometry 51F99(Primary), 15A15(Secondary) For an $n\times n$ positive definite symmetric matrix $Z$ with $Z_{ii} = 1$ for all $i$, we show that there exists a set of vectors $V_Z\subset \mathbb{R}^n$ such that the radius $R$ of the circumsphere of $V_Z$ satisfies ${\rm Mag}\ Z = (1-R^2)^{-1}$. This leads us to interpret geometrically several known and new facts on magnitude. In particular, we show that ${\rm Mag}\ Z_{X}< n$ for an $n$-point metric space $X$ of negative type with $n>1$. This result gives a negative answer to a problem given by Gomi--Meckes \cite{GM}. Furthermore, we also have a similar geometric description of magnitude for general real symmetric matrix $Z$ with $Z_{ii} = 1$ for all $i$. In this case, the radius corresponds to that of a circum-quasi-sphere, namely the set of points having a prescribed norm in a vector space endowed with an indefinite inner product. |
| title | Geometric interpretation of magnitude |
| topic | Metric Geometry 51F99(Primary), 15A15(Secondary) |
| url | https://arxiv.org/abs/2510.26118 |