Weighted cone-volume measures of pseudo-cones
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916314176552960 |
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| author | Schneider, Rolf |
| author_facet | Schneider, Rolf |
| contents | A pseudo-cone in ${\mathbb R}^n$ is a nonempty closed convex set $K$ not containing the origin and such that $λK \subseteq K$ for all $λ\ge 1$. It is called a $C$-pseudo-cone if $C$ is its recession cone, where $C$ is a pointed closed convex cone with interior points. The cone-volume measure of a pseudo-cone can be defined similarly as for convex bodies, but it may be infinite. After proving a necessary condition for cone-volume measures of $C$-pseudo-cones, we introduce suitable weights for cone-volume measures, yielding finite measures. Then we provide a necessary and sufficient condition for a Borel measure on the unit sphere to be the weighted cone-volume measure of some $C$-pseudo-cone. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05095 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weighted cone-volume measures of pseudo-cones Schneider, Rolf Metric Geometry 52A20 A pseudo-cone in ${\mathbb R}^n$ is a nonempty closed convex set $K$ not containing the origin and such that $λK \subseteq K$ for all $λ\ge 1$. It is called a $C$-pseudo-cone if $C$ is its recession cone, where $C$ is a pointed closed convex cone with interior points. The cone-volume measure of a pseudo-cone can be defined similarly as for convex bodies, but it may be infinite. After proving a necessary condition for cone-volume measures of $C$-pseudo-cones, we introduce suitable weights for cone-volume measures, yielding finite measures. Then we provide a necessary and sufficient condition for a Borel measure on the unit sphere to be the weighted cone-volume measure of some $C$-pseudo-cone. |
| title | Weighted cone-volume measures of pseudo-cones |
| topic | Metric Geometry 52A20 |
| url | https://arxiv.org/abs/2407.05095 |