Asymptotic theory of $C$-pseudo-cones
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915011747643392 |
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| author | Wang, Xudong Xu, Wenxue Zhou, Jiazu Zhu, Baocheng |
| author_facet | Wang, Xudong Xu, Wenxue Zhou, Jiazu Zhu, Baocheng |
| contents | In this paper, we study the non-degenerated $C$-pseudo-cones which can be uniquely decomposed into the sum of a $C$-asymptotic set and a $C$-starting point. Combining this with the novel work in \cite{Schneider-A_weighted_Minkowski_theorem}, we introduce the asymptotic weighted co-volume functional $T_Θ(E)$ of the non-degenerated $C$-pseudo-cone $E$, which is also a generalized function with the singular point $o$ (the origin). Using our convolution formula for $T_Θ(E)$, we establish a decay estimate for $T_Θ(E)$ at infinity and present some interesting results. As applications of this asymptotic theory, we prove a weighted Brunn-Minkowski type inequality and study the solutions to the weighted Minkowski problem for pseudo-cones. Moreover, we pose an open problem regarding $T_Θ(E)$, which we call the asymptotic Brunn-Minkowski inequality for $C$-pseudo-cones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14962 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic theory of $C$-pseudo-cones Wang, Xudong Xu, Wenxue Zhou, Jiazu Zhu, Baocheng Metric Geometry In this paper, we study the non-degenerated $C$-pseudo-cones which can be uniquely decomposed into the sum of a $C$-asymptotic set and a $C$-starting point. Combining this with the novel work in \cite{Schneider-A_weighted_Minkowski_theorem}, we introduce the asymptotic weighted co-volume functional $T_Θ(E)$ of the non-degenerated $C$-pseudo-cone $E$, which is also a generalized function with the singular point $o$ (the origin). Using our convolution formula for $T_Θ(E)$, we establish a decay estimate for $T_Θ(E)$ at infinity and present some interesting results. As applications of this asymptotic theory, we prove a weighted Brunn-Minkowski type inequality and study the solutions to the weighted Minkowski problem for pseudo-cones. Moreover, we pose an open problem regarding $T_Θ(E)$, which we call the asymptotic Brunn-Minkowski inequality for $C$-pseudo-cones. |
| title | Asymptotic theory of $C$-pseudo-cones |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2410.14962 |