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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2402.12802 |
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| _version_ | 1866913237605285888 |
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| author | Zhang, Ning |
| author_facet | Zhang, Ning |
| contents | In this paper, combining the covolume, we study the Minkowski theory for the non-compact convex set with an asymptotic boundary condition. In particular, the mixed covolume of two non-compact convex sets is introduced and its geometric interpretation is obtained by the Hadamard variational formula. The Brunn-Minkowski and Minkowski inequalities for covolume are established, and the equivalence of these two inequalities are discussed as well. The Minkowski problem for non-compact convex set is proposed and solved under the asymptotic conditions. In the end, we give a solution to the Minkowski problem for $σ$-finite measure on the conic domain $Ω_C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12802 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Minkowski problem for the non-compact convex set with an asymptotic boundary condition Zhang, Ning Differential Geometry 52B45, 52A20, 52A39, 53A15 In this paper, combining the covolume, we study the Minkowski theory for the non-compact convex set with an asymptotic boundary condition. In particular, the mixed covolume of two non-compact convex sets is introduced and its geometric interpretation is obtained by the Hadamard variational formula. The Brunn-Minkowski and Minkowski inequalities for covolume are established, and the equivalence of these two inequalities are discussed as well. The Minkowski problem for non-compact convex set is proposed and solved under the asymptotic conditions. In the end, we give a solution to the Minkowski problem for $σ$-finite measure on the conic domain $Ω_C$. |
| title | The Minkowski problem for the non-compact convex set with an asymptotic boundary condition |
| topic | Differential Geometry 52B45, 52A20, 52A39, 53A15 |
| url | https://arxiv.org/abs/2402.12802 |