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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.05913 |
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| _version_ | 1866914638087585792 |
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| author | Colesanti, Andrea Knoerr, Jonas Pagnini, Daniele |
| author_facet | Colesanti, Andrea Knoerr, Jonas Pagnini, Daniele |
| contents | We show that every continuous and dually translation invariant valuation on the space of Lipschitz functions on the unit sphere of $\mathbb{R}^n$, $n\ge2$, can be decomposed uniquely into a sum of homogeneous valuations of degree $0$, $1$ and $2$. In particular, there does not exist any non-trivial, continuous and dually translation invariant valuation which is homogeneous of degree $3$ or higher. For the space of those of degree $0$, $1$ and $2$ we provide a description of a dense subspace. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05913 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The homogeneous decomposition of dually translation invariant valuations on Lipschitz functions on the sphere Colesanti, Andrea Knoerr, Jonas Pagnini, Daniele Metric Geometry Functional Analysis 52B45, 26A16 We show that every continuous and dually translation invariant valuation on the space of Lipschitz functions on the unit sphere of $\mathbb{R}^n$, $n\ge2$, can be decomposed uniquely into a sum of homogeneous valuations of degree $0$, $1$ and $2$. In particular, there does not exist any non-trivial, continuous and dually translation invariant valuation which is homogeneous of degree $3$ or higher. For the space of those of degree $0$, $1$ and $2$ we provide a description of a dense subspace. |
| title | The homogeneous decomposition of dually translation invariant valuations on Lipschitz functions on the sphere |
| topic | Metric Geometry Functional Analysis 52B45, 26A16 |
| url | https://arxiv.org/abs/2401.05913 |