Hyperbolic embedding of infinite-dimensional convex bodies
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914957388414976 |
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| author | Long, Yusen |
| author_facet | Long, Yusen |
| contents | In this article, we use the second intrinsic volume to define a metric on the space of homothetic classes of Gaussian bounded convex bodies in a separable real Hilbert space. Using kernels of hyperbolic type, we can deduce that this space is isometrically embedded into an infinite-dimensional real hyperbolic space. Applying Malliavin calculus, it is possible to adapt integral geometry for convex bodies in infinite dimensions. Moreover, we give a new formula for computing second intrinsic volumes of convex bodies and offer a description of the completion for the hyperbolic embedding of Gaussian bounded convex bodies with dimension at least two and thus answer a question asked by Debin and Fillastre [DF22]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_13428 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hyperbolic embedding of infinite-dimensional convex bodies Long, Yusen Metric Geometry Probability 52A07 (primary), 60D05 (secondary) In this article, we use the second intrinsic volume to define a metric on the space of homothetic classes of Gaussian bounded convex bodies in a separable real Hilbert space. Using kernels of hyperbolic type, we can deduce that this space is isometrically embedded into an infinite-dimensional real hyperbolic space. Applying Malliavin calculus, it is possible to adapt integral geometry for convex bodies in infinite dimensions. Moreover, we give a new formula for computing second intrinsic volumes of convex bodies and offer a description of the completion for the hyperbolic embedding of Gaussian bounded convex bodies with dimension at least two and thus answer a question asked by Debin and Fillastre [DF22]. |
| title | Hyperbolic embedding of infinite-dimensional convex bodies |
| topic | Metric Geometry Probability 52A07 (primary), 60D05 (secondary) |
| url | https://arxiv.org/abs/2305.13428 |