On the discrete logarithmic Minkowski problem in the plane
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866909995235278848 |
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| author | Baumbach, Tom |
| author_facet | Baumbach, Tom |
| contents | The paper characterizes the convex hull of the closure of the cone-volume set $C_\cv(U)$, consisting of all cone-volume vectors of polygons with outer unit normals vectors contained in $U$, for any finite set $U \subseteq \R^2, \pos(U) = \R^2$. We prove that this convex hull has finitely many extreme points by providing both a vertex representation as well as a half space representation. As a consequence, we derive new necessary conditions, which depend on $U$, for the existence of solutions to the logarithmic Minkowski problem in $\R^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13159 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the discrete logarithmic Minkowski problem in the plane Baumbach, Tom Metric Geometry 52A10, 52B05 The paper characterizes the convex hull of the closure of the cone-volume set $C_\cv(U)$, consisting of all cone-volume vectors of polygons with outer unit normals vectors contained in $U$, for any finite set $U \subseteq \R^2, \pos(U) = \R^2$. We prove that this convex hull has finitely many extreme points by providing both a vertex representation as well as a half space representation. As a consequence, we derive new necessary conditions, which depend on $U$, for the existence of solutions to the logarithmic Minkowski problem in $\R^2$. |
| title | On the discrete logarithmic Minkowski problem in the plane |
| topic | Metric Geometry 52A10, 52B05 |
| url | https://arxiv.org/abs/2601.13159 |