Smooth valuations on convex bodies and finite linear combinations of mixed volumes
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914971990884352 |
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| author | Knoerr, Jonas |
| author_facet | Knoerr, Jonas |
| contents | It is shown that Alesker's solution of McMullen's conjecture implies the following stronger version of the conjecture: Every continuous, translation invariant, $k$-homogeneous valuation on convex bodies in $\mathbb{R}^n$ can be approximated uniformly on compact subsets by finite linear combinations of mixed volumes involving at most $N_{n,k}$ summands, where $N_{n,k}$ is a constant depending on $n$ and $k$ only. Moreover, $n-k-1$ of the arguments of the mixed volumes can be chosen to be ellipsoids that do not depend on the valuation. The result is based on a corresponding description of smooth valuations in terms of finite linear combinations of mixed volumes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08183 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Smooth valuations on convex bodies and finite linear combinations of mixed volumes Knoerr, Jonas Metric Geometry 52B45, 52A20, 52A39 It is shown that Alesker's solution of McMullen's conjecture implies the following stronger version of the conjecture: Every continuous, translation invariant, $k$-homogeneous valuation on convex bodies in $\mathbb{R}^n$ can be approximated uniformly on compact subsets by finite linear combinations of mixed volumes involving at most $N_{n,k}$ summands, where $N_{n,k}$ is a constant depending on $n$ and $k$ only. Moreover, $n-k-1$ of the arguments of the mixed volumes can be chosen to be ellipsoids that do not depend on the valuation. The result is based on a corresponding description of smooth valuations in terms of finite linear combinations of mixed volumes. |
| title | Smooth valuations on convex bodies and finite linear combinations of mixed volumes |
| topic | Metric Geometry 52B45, 52A20, 52A39 |
| url | https://arxiv.org/abs/2312.08183 |